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A study of 24
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Games & mathematics

Seven Ways to See 24

What makes a 24 puzzle difficult? I classified all 1,362 solvable hands by their easiest solution, from familiar factors to fractions hiding in plain sight.

In Target Number, the rules for 24 are simple: take four numbers, use each once, and combine them with addition, subtraction, multiplication, and division to make 24. The arithmetic is usually manageable. Seeing what to do with the numbers is the interesting part.

I started wondering whether that act of recognition could be organized. Finding 6 × 4 seems easier than finding 35 − 11, even though both finish at the same place. And a hand that asks you to construct two differences has something in common with other hands that use the same pattern.

So I worked through all 1,820 distinct four-number combinations from 1 through 13, allowing repeats. Of those, 1,362 can make 24. I then classified each solvable hand by the simplest kind of solution available to it, using a proposed hierarchy of seven levels.

One hand, three very different ideas

Before ranking anything, there is a trap to avoid: the first solution you find may make a puzzle look much harder than it is. Take these four numbers.

2589
A larger multiple

(9 × 8) ÷ (5 − 2) = 24

Two familiar factors

(9 − 5) × (8 − 2) = 24

Or, just add

2 + 5 + 8 + 9 = 24

The first route builds 72 and divides by 3. The second builds 4 × 6. The third simply adds the numbers. Under this hierarchy, those are Level 5, Level 3, and Level 1 solutions to the same hand. The hand gets Level 1.

That is the central rule of the analysis: rank the solutions first, then give each hand its lowest available level. Every solution uses three operations, so counting operations would not distinguish any of these puzzles.

The seven levels

This is my proposed order for teaching the patterns. It is a hypothesis about what is easier to recognize, rather than a ranking measured from players' solve times. Try the examples below before revealing their solutions. Each hand's easiest solution falls at the level shown.

1Direct arithmetic

Start with the simplest check: do the numbers already add to 24? This level also includes routes using only addition and subtraction, or only multiplication, without negative intermediate values.

24711
Reveal solutionHide solution for level 1

2 + 4 + 7 + 11 = 24

No factor needs to be constructed. The four numbers already make the total.

Browse all 226 problemsHide 226 problems · Level 1

Ordered by the four numbers. Select a hand to reveal one of its easiest solutions. Scroll the list to see every problem in this level.

  • 11212Solution

    ((12 × 2) × 1) × 1 = 24

  • 1138Solution

    8 × (3 × (1 × 1)) = 24

  • 1146Solution

    6 × (4 × (1 × 1)) = 24

  • 11913Solution

    (13 + (1 + 1)) + 9 = 24

  • 111012Solution

    (10 + (1 + 1)) + 12 = 24

  • 111111Solution

    (11 + (1 + 1)) + 11 = 24

  • 111113Solution

    13 + ((11 + 1) − 1) = 24

  • 111212Solution

    ((12 + 1) − 1) + 12 = 24

  • 111313Solution

    13 + (13 − (1 + 1)) = 24

  • 1226Solution

    6 × ((2 × 1) × 2) = 24

  • 1234Solution

    (3 × (2 × 1)) × 4 = 24

  • 12813Solution

    ((13 + 1) + 2) + 8 = 24

  • 12912Solution

    ((12 + 1) + 2) + 9 = 24

  • 121011Solution

    ((10 + 1) + 11) + 2 = 24

  • 121013Solution

    ((13 + 10) − 1) + 2 = 24

  • 121112Solution

    ((12 + 11) − 1) + 2 = 24

  • 121213Solution

    13 + ((12 + 1) − 2) = 24

  • 13713Solution

    ((13 + 1) + 3) + 7 = 24

  • 13812Solution

    ((12 + 1) + 3) + 8 = 24

  • 13911Solution

    ((11 + 1) + 3) + 9 = 24

  • 13913Solution

    ((13 + 3) − 1) + 9 = 24

  • 131010Solution

    ((10 + 1) + 10) + 3 = 24

  • 131012Solution

    ((12 + 10) − 1) + 3 = 24

  • 131111Solution

    ((11 + 11) − 1) + 3 = 24

  • 131313Solution

    13 + ((13 + 1) − 3) = 24

  • 14613Solution

    ((13 + 1) + 4) + 6 = 24

  • 14712Solution

    ((12 + 1) + 4) + 7 = 24

  • 14811Solution

    ((11 + 1) + 4) + 8 = 24

  • 14813Solution

    ((13 + 4) − 1) + 8 = 24

  • 14910Solution

    ((10 + 1) + 4) + 9 = 24

  • 14912Solution

    ((12 + 4) − 1) + 9 = 24

  • 141011Solution

    ((11 + 10) − 1) + 4 = 24

  • 15513Solution

    ((13 + 1) + 5) + 5 = 24

  • 15612Solution

    ((12 + 1) + 5) + 6 = 24

  • 15711Solution

    ((11 + 1) + 5) + 7 = 24

  • 15713Solution

    ((13 + 5) − 1) + 7 = 24

  • 15810Solution

    ((10 + 1) + 5) + 8 = 24

  • 15812Solution

    ((12 + 5) − 1) + 8 = 24

  • 1599Solution

    (9 + (5 + 1)) + 9 = 24

  • 15911Solution

    ((11 + 5) − 1) + 9 = 24

  • 151010Solution

    ((10 + 10) − 1) + 5 = 24

  • 16611Solution

    ((11 + 1) + 6) + 6 = 24

  • 16613Solution

    ((13 + 6) − 1) + 6 = 24

  • 16710Solution

    ((10 + 1) + 6) + 7 = 24

  • 16712Solution

    ((12 + 6) − 1) + 7 = 24

  • 1689Solution

    (8 + (6 + 1)) + 9 = 24

  • 16811Solution

    ((11 + 6) − 1) + 8 = 24

  • 16910Solution

    ((10 + 6) − 1) + 9 = 24

  • 1779Solution

    ((7 + 1) + 7) + 9 = 24

  • 17711Solution

    ((11 + 7) − 1) + 7 = 24

  • 1788Solution

    ((7 + 1) + 8) + 8 = 24

  • 17810Solution

    ((10 + 7) − 1) + 8 = 24

  • 1799Solution

    ((9 + 7) − 1) + 9 = 24

  • 1889Solution

    ((8 + 8) − 1) + 9 = 24

  • 2223Solution

    ((2 × 2) × 2) × 3 = 24

  • 22713Solution

    ((13 + 2) + 2) + 7 = 24

  • 22812Solution

    ((12 + 2) + 2) + 8 = 24

  • 22911Solution

    ((11 + 2) + 2) + 9 = 24

  • 221010Solution

    ((10 + 10) + 2) + 2 = 24

  • 221113Solution

    ((13 + 11) − 2) + 2 = 24

  • 221212Solution

    ((12 + 12) − 2) + 2 = 24

  • 23613Solution

    ((13 + 2) + 3) + 6 = 24

  • 23712Solution

    ((12 + 2) + 3) + 7 = 24

  • 23811Solution

    ((11 + 2) + 3) + 8 = 24

  • 23910Solution

    ((10 + 2) + 3) + 9 = 24

  • 231013Solution

    ((13 + 10) − 2) + 3 = 24

  • 231112Solution

    ((12 + 11) − 2) + 3 = 24

  • 231213Solution

    13 + ((12 + 2) − 3) = 24

  • 24513Solution

    ((13 + 2) + 4) + 5 = 24

  • 24612Solution

    ((12 + 2) + 4) + 6 = 24

  • 24711Solution

    ((11 + 2) + 4) + 7 = 24

  • 24810Solution

    ((10 + 2) + 4) + 8 = 24

  • 2499Solution

    (9 + (4 + 2)) + 9 = 24

  • 24913Solution

    ((13 + 4) − 2) + 9 = 24

  • 241012Solution

    ((12 + 10) − 2) + 4 = 24

  • 241111Solution

    ((11 + 11) − 2) + 4 = 24

  • 241313Solution

    13 + ((13 + 2) − 4) = 24

  • 25512Solution

    ((12 + 2) + 5) + 5 = 24

  • 25611Solution

    ((11 + 2) + 5) + 6 = 24

  • 25710Solution

    ((10 + 2) + 5) + 7 = 24

  • 2589Solution

    (8 + (5 + 2)) + 9 = 24

  • 25813Solution

    ((13 + 5) − 2) + 8 = 24

  • 25912Solution

    ((12 + 5) − 2) + 9 = 24

  • 251011Solution

    ((11 + 10) − 2) + 5 = 24

  • 26610Solution

    ((10 + 2) + 6) + 6 = 24

  • 2679Solution

    ((6 + 2) + 7) + 9 = 24

  • 26713Solution

    ((13 + 6) − 2) + 7 = 24

  • 2688Solution

    ((6 + 2) + 8) + 8 = 24

  • 26812Solution

    ((12 + 6) − 2) + 8 = 24

  • 26911Solution

    ((11 + 6) − 2) + 9 = 24

  • 261010Solution

    ((10 + 10) − 2) + 6 = 24

  • 2778Solution

    ((7 + 2) + 7) + 8 = 24

  • 27712Solution

    ((12 + 7) − 2) + 7 = 24

  • 27811Solution

    ((11 + 7) − 2) + 8 = 24

  • 27910Solution

    ((10 + 7) − 2) + 9 = 24

  • 28810Solution

    ((10 + 8) − 2) + 8 = 24

  • 2899Solution

    ((9 + 8) − 2) + 9 = 24

  • 33513Solution

    ((13 + 3) + 3) + 5 = 24

  • 33612Solution

    ((12 + 3) + 3) + 6 = 24

  • 33711Solution

    ((11 + 3) + 3) + 7 = 24

  • 33810Solution

    ((10 + 3) + 3) + 8 = 24

  • 3399Solution

    (9 + (3 + 3)) + 9 = 24

  • 331113Solution

    ((13 + 11) − 3) + 3 = 24

  • 331212Solution

    ((12 + 12) − 3) + 3 = 24

  • 34413Solution

    ((13 + 3) + 4) + 4 = 24

  • 34512Solution

    ((12 + 3) + 4) + 5 = 24

  • 34611Solution

    ((11 + 3) + 4) + 6 = 24

  • 34710Solution

    ((10 + 3) + 4) + 7 = 24

  • 3489Solution

    (8 + (4 + 3)) + 9 = 24

  • 341013Solution

    ((13 + 10) − 3) + 4 = 24

  • 341112Solution

    ((12 + 11) − 3) + 4 = 24

  • 341213Solution

    13 + ((12 + 3) − 4) = 24

  • 35511Solution

    ((11 + 3) + 5) + 5 = 24

  • 35610Solution

    ((10 + 3) + 5) + 6 = 24

  • 3579Solution

    ((5 + 3) + 7) + 9 = 24

  • 3588Solution

    ((5 + 3) + 8) + 8 = 24

  • 35913Solution

    ((13 + 5) − 3) + 9 = 24

  • 351012Solution

    ((12 + 10) − 3) + 5 = 24

  • 351111Solution

    ((11 + 11) − 3) + 5 = 24

  • 351313Solution

    13 + ((13 + 3) − 5) = 24

  • 3669Solution

    ((6 + 3) + 6) + 9 = 24

  • 3678Solution

    ((6 + 3) + 7) + 8 = 24

  • 36813Solution

    ((13 + 6) − 3) + 8 = 24

  • 36912Solution

    ((12 + 6) − 3) + 9 = 24

  • 361011Solution

    ((11 + 10) − 3) + 6 = 24

  • 3777Solution

    ((7 + 3) + 7) + 7 = 24

  • 37713Solution

    ((13 + 7) − 3) + 7 = 24

  • 37812Solution

    ((12 + 7) − 3) + 8 = 24

  • 37911Solution

    ((11 + 7) − 3) + 9 = 24

  • 371010Solution

    ((10 + 10) − 3) + 7 = 24

  • 38811Solution

    ((11 + 8) − 3) + 8 = 24

  • 38910Solution

    ((10 + 8) − 3) + 9 = 24

  • 3999Solution

    ((9 + 9) − 3) + 9 = 24

  • 44412Solution

    ((12 + 4) + 4) + 4 = 24

  • 44511Solution

    ((11 + 4) + 4) + 5 = 24

  • 44610Solution

    ((10 + 4) + 4) + 6 = 24

  • 4479Solution

    ((4 + 4) + 7) + 9 = 24

  • 4488Solution

    ((4 + 4) + 8) + 8 = 24

  • 441113Solution

    ((13 + 11) − 4) + 4 = 24

  • 441212Solution

    ((12 + 12) − 4) + 4 = 24

  • 45510Solution

    ((10 + 4) + 5) + 5 = 24

  • 4569Solution

    ((5 + 4) + 6) + 9 = 24

  • 4578Solution

    ((5 + 4) + 7) + 8 = 24

  • 451013Solution

    ((13 + 10) − 4) + 5 = 24

  • 451112Solution

    ((12 + 11) − 4) + 5 = 24

  • 451213Solution

    13 + ((12 + 4) − 5) = 24

  • 4668Solution

    ((6 + 4) + 6) + 8 = 24

  • 4677Solution

    ((6 + 4) + 7) + 7 = 24

  • 46913Solution

    ((13 + 6) − 4) + 9 = 24

  • 461012Solution

    ((12 + 10) − 4) + 6 = 24

  • 461111Solution

    ((11 + 11) − 4) + 6 = 24

  • 461313Solution

    13 + ((13 + 4) − 6) = 24

  • 47813Solution

    ((13 + 7) − 4) + 8 = 24

  • 47912Solution

    ((12 + 7) − 4) + 9 = 24

  • 471011Solution

    ((11 + 10) − 4) + 7 = 24

  • 48812Solution

    ((12 + 8) − 4) + 8 = 24

  • 48911Solution

    ((11 + 8) − 4) + 9 = 24

  • 481010Solution

    ((10 + 10) − 4) + 8 = 24

  • 49910Solution

    ((10 + 9) − 4) + 9 = 24

  • 5559Solution

    ((5 + 5) + 5) + 9 = 24

  • 5568Solution

    ((5 + 5) + 6) + 8 = 24

  • 5577Solution

    ((5 + 5) + 7) + 7 = 24

  • 551113Solution

    ((13 + 11) − 5) + 5 = 24

  • 551212Solution

    ((12 + 12) − 5) + 5 = 24

  • 5667Solution

    ((6 + 5) + 6) + 7 = 24

  • 561013Solution

    ((13 + 10) − 5) + 6 = 24

  • 561112Solution

    ((12 + 11) − 5) + 6 = 24

  • 561213Solution

    13 + ((12 + 5) − 6) = 24

  • 57913Solution

    ((13 + 7) − 5) + 9 = 24

  • 571012Solution

    ((12 + 10) − 5) + 7 = 24

  • 571111Solution

    ((11 + 11) − 5) + 7 = 24

  • 571313Solution

    13 + ((13 + 5) − 7) = 24

  • 58813Solution

    ((13 + 8) − 5) + 8 = 24

  • 58912Solution

    ((12 + 8) − 5) + 9 = 24

  • 581011Solution

    ((11 + 10) − 5) + 8 = 24

  • 59911Solution

    ((11 + 9) − 5) + 9 = 24

  • 591010Solution

    ((10 + 10) − 5) + 9 = 24

  • 6666Solution

    ((6 + 6) + 6) + 6 = 24

  • 661113Solution

    ((13 + 11) − 6) + 6 = 24

  • 661212Solution

    ((12 + 12) − 6) + 6 = 24

  • 671013Solution

    ((13 + 10) − 6) + 7 = 24

  • 671112Solution

    ((12 + 11) − 6) + 7 = 24

  • 671213Solution

    13 + ((12 + 6) − 7) = 24

  • 68913Solution

    ((13 + 8) − 6) + 9 = 24

  • 681012Solution

    ((12 + 10) − 6) + 8 = 24

  • 681111Solution

    ((11 + 11) − 6) + 8 = 24

  • 681313Solution

    13 + ((13 + 6) − 8) = 24

  • 69912Solution

    ((12 + 9) − 6) + 9 = 24

  • 691011Solution

    ((11 + 10) − 6) + 9 = 24

  • 6101010Solution

    ((10 + 10) − 6) + 10 = 24

  • 771113Solution

    ((13 + 11) − 7) + 7 = 24

  • 771212Solution

    ((12 + 12) − 7) + 7 = 24

  • 781013Solution

    ((13 + 10) − 7) + 8 = 24

  • 781112Solution

    ((12 + 11) − 7) + 8 = 24

  • 781213Solution

    13 + ((12 + 7) − 8) = 24

  • 79913Solution

    ((13 + 9) − 7) + 9 = 24

  • 791012Solution

    ((12 + 10) − 7) + 9 = 24

  • 791111Solution

    ((11 + 11) − 7) + 9 = 24

  • 791313Solution

    13 + ((13 + 7) − 9) = 24

  • 7101011Solution

    ((10 + 10) − 7) + 11 = 24

  • 881113Solution

    ((13 + 11) − 8) + 8 = 24

  • 881212Solution

    ((12 + 12) − 8) + 8 = 24

  • 891013Solution

    ((13 + 10) − 8) + 9 = 24

  • 891112Solution

    ((12 + 11) − 8) + 9 = 24

  • 891213Solution

    13 + ((12 + 8) − 9) = 24

  • 8101012Solution

    ((10 + 10) − 8) + 12 = 24

  • 8101111Solution

    ((11 + 10) − 8) + 11 = 24

  • 8101313Solution

    13 + ((13 + 8) − 10) = 24

  • 991113Solution

    ((13 + 11) − 9) + 9 = 24

  • 991212Solution

    ((12 + 12) − 9) + 9 = 24

  • 9101013Solution

    13 + ((10 + 10) − 9) = 24

  • 9101112Solution

    ((11 + 10) − 9) + 12 = 24

  • 9101213Solution

    13 + ((12 + 9) − 10) = 24

  • 9111111Solution

    ((11 + 11) − 9) + 11 = 24

  • 9111313Solution

    13 + ((13 + 9) − 11) = 24

  • 10101113Solution

    13 + ((11 + 10) − 10) = 24

  • 10101212Solution

    ((12 + 10) − 10) + 12 = 24

  • 10111112Solution

    ((11 + 11) − 10) + 12 = 24

  • 10111213Solution

    13 + ((12 + 10) − 11) = 24

  • 10121313Solution

    13 + ((13 + 10) − 12) = 24

  • 11111113Solution

    13 + ((11 + 11) − 11) = 24

  • 11111212Solution

    ((12 + 11) − 11) + 12 = 24

  • 11121213Solution

    13 + ((12 + 11) − 12) = 24

  • 11131313Solution

    13 + ((13 + 11) − 13) = 24

  • 12121212Solution

    ((12 + 12) − 12) + 12 = 24

  • 12121313Solution

    13 + ((12 + 12) − 13) = 24

2Obvious factors and cancellation

Sometimes the useful numbers are already there. The job is to make the other cards harmless, often by turning a matching pair into zero or one. Simple exact division and a single preparatory step followed by multiplication also belong here.

4699
Reveal solutionHide solution for level 2

4 × 6 × (9 ÷ 9) = 24

The 4 and 6 do the work; the two 9s make 1.

Browse all 226 problemsHide 226 problems · Level 2

Ordered by the four numbers. Select a hand to reveal one of its easiest solutions. Scroll the list to see every problem in this level.

  • 11112Solution

    12 × ((1 + 1) × 1) = 24

  • 1126Solution

    6 × ((1 + 1) × 2) = 24

  • 1128Solution

    8 × ((2 + 1) × 1) = 24

  • 11211Solution

    ((11 + 1) × 1) × 2 = 24

  • 11213Solution

    ((13 − 1) × 1) × 2 = 24

  • 1134Solution

    (3 × (1 + 1)) × 4 = 24

  • 1136Solution

    6 × ((3 + 1) × 1) = 24

  • 1137Solution

    ((7 + 1) × 1) × 3 = 24

  • 1139Solution

    ((9 × 1) − 1) × 3 = 24

  • 11312Solution

    12 × ((3 × 1) − 1) = 24

  • 1145Solution

    ((5 + 1) × 1) × 4 = 24

  • 1147Solution

    ((7 × 1) − 1) × 4 = 24

  • 1148Solution

    8 × ((4 × 1) − 1) = 24

  • 1156Solution

    6 × ((5 × 1) − 1) = 24

  • 111013Solution

    (13 + (10 + 1)) × 1 = 24

  • 111112Solution

    ((11 + 1) + 12) × 1 = 24

  • 111213Solution

    (13 + (12 − 1)) × 1 = 24

  • 1224Solution

    ((2 + 1) × 2) × 4 = 24

  • 1225Solution

    ((5 + 1) × 2) × 2 = 24

  • 1227Solution

    ((7 − 1) × 2) × 2 = 24

  • 12210Solution

    ((10 + 2) × 1) × 2 = 24

  • 12212Solution

    (12 × (2 − 1)) × 2 = 24

  • 1233Solution

    ((3 + 1) × 2) × 3 = 24

  • 1235Solution

    ((5 − 1) × 2) × 3 = 24

  • 1236Solution

    (6 + (2 × 1)) × 3 = 24

  • 1238Solution

    8 × (3 × (2 − 1)) = 24

  • 1239Solution

    (9 + (3 × 1)) × 2 = 24

  • 12310Solution

    ((10 − 2) × 1) × 3 = 24

  • 1244Solution

    (4 + (2 × 1)) × 4 = 24

  • 1246Solution

    6 × (4 × (2 − 1)) = 24

  • 1248Solution

    (8 + (4 × 1)) × 2 = 24

  • 12412Solution

    12 × ((4 × 1) − 2) = 24

  • 1257Solution

    (7 + (5 × 1)) × 2 = 24

  • 1258Solution

    8 × ((5 × 1) − 2) = 24

  • 1266Solution

    ((6 × 1) + 6) × 2 = 24

  • 1268Solution

    8 × ((6 × 1) ÷ 2) = 24

  • 12913Solution

    ((13 + 2) + 9) × 1 = 24

  • 121012Solution

    ((12 + 10) + 2) × 1 = 24

  • 121111Solution

    ((11 + 11) + 2) × 1 = 24

  • 121113Solution

    13 + (11 × (2 − 1)) = 24

  • 121212Solution

    (12 × (2 − 1)) + 12 = 24

  • 121313Solution

    (13 + (13 − 2)) × 1 = 24

  • 1334Solution

    ((3 × 1) + 3) × 4 = 24

  • 1335Solution

    (5 + (3 × 1)) × 3 = 24

  • 13311Solution

    ((11 − 3) × 1) × 3 = 24

  • 1344Solution

    ((4 × 1) + 4) × 3 = 24

  • 1349Solution

    ((9 × 1) − 3) × 4 = 24

  • 13412Solution

    ((12 − 4) × 1) × 3 = 24

  • 13512Solution

    12 × ((5 × 1) − 3) = 24

  • 13513Solution

    ((13 − 5) × 1) × 3 = 24

  • 1367Solution

    6 × ((7 × 1) − 3) = 24

  • 1368Solution

    8 × ((6 × 1) − 3) = 24

  • 13612Solution

    6 × ((12 ÷ 3) × 1) = 24

  • 1389Solution

    8 × ((9 × 1) ÷ 3) = 24

  • 13813Solution

    ((13 + 3) + 8) × 1 = 24

  • 13912Solution

    ((12 + 3) + 9) × 1 = 24

  • 131011Solution

    ((11 + 10) + 3) × 1 = 24

  • 14410Solution

    ((10 − 4) × 1) × 4 = 24

  • 14511Solution

    ((11 − 5) × 1) × 4 = 24

  • 1468Solution

    6 × ((8 × 1) − 4) = 24

  • 14612Solution

    ((12 − 6) × 1) × 4 = 24

  • 1478Solution

    8 × ((7 × 1) − 4) = 24

  • 14713Solution

    ((13 + 4) + 7) × 1 = 24

  • 14812Solution

    ((12 + 4) + 8) × 1 = 24

  • 14911Solution

    ((11 + 4) + 9) × 1 = 24

  • 141010Solution

    ((10 + 10) + 4) × 1 = 24

  • 1569Solution

    6 × ((9 × 1) − 5) = 24

  • 15613Solution

    ((13 + 5) + 6) × 1 = 24

  • 15712Solution

    ((12 + 5) + 7) × 1 = 24

  • 1588Solution

    8 × ((8 × 1) − 5) = 24

  • 15811Solution

    ((11 + 5) + 8) × 1 = 24

  • 15910Solution

    ((10 + 5) + 9) × 1 = 24

  • 151012Solution

    12 × ((10 ÷ 5) × 1) = 24

  • 16610Solution

    6 × ((10 − 6) × 1) = 24

  • 16612Solution

    ((12 + 6) + 6) × 1 = 24

  • 16711Solution

    ((11 + 6) + 7) × 1 = 24

  • 16810Solution

    ((10 + 6) + 8) × 1 = 24

  • 16812Solution

    6 × ((12 − 8) × 1) = 24

  • 1699Solution

    (9 + (6 × 1)) + 9 = 24

  • 16913Solution

    6 × ((13 − 9) × 1) = 24

  • 161212Solution

    12 × ((12 ÷ 6) × 1) = 24

  • 17710Solution

    ((10 + 7) + 7) × 1 = 24

  • 1789Solution

    (8 + (7 × 1)) + 9 = 24

  • 17912Solution

    12 × ((9 × 1) − 7) = 24

  • 1888Solution

    ((8 × 1) + 8) + 8 = 24

  • 18811Solution

    8 × ((11 − 8) × 1) = 24

  • 18912Solution

    8 × ((12 − 9) × 1) = 24

  • 181012Solution

    12 × ((10 − 8) × 1) = 24

  • 181013Solution

    8 × ((13 − 10) × 1) = 24

  • 191112Solution

    12 × ((11 − 9) × 1) = 24

  • 1101212Solution

    12 × ((12 − 10) × 1) = 24

  • 1111213Solution

    12 × ((13 − 11) × 1) = 24

  • 2224Solution

    ((4 + 2) × 2) × 2 = 24

  • 2228Solution

    ((8 − 2) × 2) × 2 = 24

  • 22212Solution

    (12 + (2 − 2)) × 2 = 24

  • 2233Solution

    ((3 + 3) × 2) × 2 = 24

  • 2236Solution

    ((6 − 2) × 2) × 3 = 24

  • 2238Solution

    8 × (3 + (2 − 2)) = 24

  • 2239Solution

    ((9 − 3) × 2) × 2 = 24

  • 22312Solution

    (12 × (3 − 2)) × 2 = 24

  • 2245Solution

    ((5 − 2) × 2) × 4 = 24

  • 2246Solution

    6 × (4 + (2 − 2)) = 24

  • 22410Solution

    ((10 − 4) × 2) × 2 = 24

  • 22511Solution

    ((11 − 5) × 2) × 2 = 24

  • 22612Solution

    ((12 − 6) × 2) × 2 = 24

  • 2337Solution

    ((7 − 3) × 2) × 3 = 24

  • 2338Solution

    8 × (3 × (3 − 2)) = 24

  • 23312Solution

    (12 + (3 − 3)) × 2 = 24

  • 2344Solution

    (3 × (4 − 2)) × 4 = 24

  • 2346Solution

    6 × (4 × (3 − 2)) = 24

  • 2348Solution

    ((8 − 4) × 2) × 3 = 24

  • 2349Solution

    ((9 × 2) ÷ 3) × 4 = 24

  • 23412Solution

    (12 × (4 − 3)) × 2 = 24

  • 2356Solution

    6 × ((5 − 3) × 2) = 24

  • 2359Solution

    ((9 − 5) × 2) × 3 = 24

  • 2366Solution

    6 × ((6 × 2) ÷ 3) = 24

  • 23610Solution

    ((10 − 6) × 2) × 3 = 24

  • 23711Solution

    ((11 − 7) × 2) × 3 = 24

  • 23812Solution

    ((12 − 8) × 2) × 3 = 24

  • 23913Solution

    ((13 − 9) × 2) × 3 = 24

  • 231113Solution

    13 + (11 × (3 − 2)) = 24

  • 231212Solution

    (12 × (3 − 2)) + 12 = 24

  • 2447Solution

    ((7 − 4) × 2) × 4 = 24

  • 24412Solution

    (12 + (4 − 4)) × 2 = 24

  • 2458Solution

    ((8 − 5) × 2) × 4 = 24

  • 24512Solution

    (12 × (5 − 4)) × 2 = 24

  • 2466Solution

    6 × ((6 − 4) × 2) = 24

  • 2468Solution

    8 × ((6 × 2) ÷ 4) = 24

  • 2469Solution

    ((9 − 6) × 2) × 4 = 24

  • 24710Solution

    ((10 − 7) × 2) × 4 = 24

  • 24811Solution

    ((11 − 8) × 2) × 4 = 24

  • 24912Solution

    ((12 − 9) × 2) × 4 = 24

  • 241013Solution

    ((13 − 10) × 2) × 4 = 24

  • 2567Solution

    6 × ((7 − 5) × 2) = 24

  • 25610Solution

    6 × ((10 × 2) ÷ 5) = 24

  • 25612Solution

    (12 × (6 − 5)) × 2 = 24

  • 2668Solution

    6 × ((8 − 6) × 2) = 24

  • 26612Solution

    (12 + (6 − 6)) × 2 = 24

  • 26712Solution

    (12 × (7 − 6)) × 2 = 24

  • 2689Solution

    8 × ((9 × 2) ÷ 6) = 24

  • 26810Solution

    6 × ((10 − 8) × 2) = 24

  • 261012Solution

    6 × ((12 − 10) × 2) = 24

  • 261113Solution

    6 × ((13 − 11) × 2) = 24

  • 27812Solution

    (12 × (8 − 7)) × 2 = 24

  • 28812Solution

    (12 + (8 − 8)) × 2 = 24

  • 28912Solution

    (12 × (9 − 8)) × 2 = 24

  • 29912Solution

    (12 + (9 − 9)) × 2 = 24

  • 291012Solution

    (12 × (10 − 9)) × 2 = 24

  • 2101012Solution

    (12 + (10 − 10)) × 2 = 24

  • 2101112Solution

    (12 × (11 − 10)) × 2 = 24

  • 2111112Solution

    (12 + (11 − 11)) × 2 = 24

  • 2111212Solution

    (12 × (12 − 11)) × 2 = 24

  • 2121212Solution

    (12 + (12 − 12)) × 2 = 24

  • 2121213Solution

    (12 × (13 − 12)) × 2 = 24

  • 2121313Solution

    (12 + (13 − 13)) × 2 = 24

  • 3338Solution

    8 × (3 + (3 − 3)) = 24

  • 3345Solution

    (3 × (5 − 3)) × 4 = 24

  • 3346Solution

    6 × (4 + (3 − 3)) = 24

  • 3348Solution

    8 × (3 × (4 − 3)) = 24

  • 3446Solution

    6 × (4 × (4 − 3)) = 24

  • 3448Solution

    8 × (3 + (4 − 4)) = 24

  • 3457Solution

    (3 × (7 − 5)) × 4 = 24

  • 3458Solution

    8 × (3 × (5 − 4)) = 24

  • 34510Solution

    (3 × (10 ÷ 5)) × 4 = 24

  • 3468Solution

    (3 × (8 − 6)) × 4 = 24

  • 34612Solution

    (3 × (12 ÷ 6)) × 4 = 24

  • 3479Solution

    (3 × (9 − 7)) × 4 = 24

  • 34810Solution

    (3 × (10 − 8)) × 4 = 24

  • 34911Solution

    (3 × (11 − 9)) × 4 = 24

  • 341012Solution

    (3 × (12 − 10)) × 4 = 24

  • 341113Solution

    (3 × (13 − 11)) × 4 = 24

  • 341212Solution

    (12 × (4 − 3)) + 12 = 24

  • 3558Solution

    8 × (3 + (5 − 5)) = 24

  • 3568Solution

    8 × (3 × (6 − 5)) = 24

  • 3668Solution

    8 × (3 + (6 − 6)) = 24

  • 3778Solution

    8 × (3 + (7 − 7)) = 24

  • 3788Solution

    8 × (3 × (8 − 7)) = 24

  • 3888Solution

    8 × (3 + (8 − 8)) = 24

  • 3889Solution

    8 × (3 × (9 − 8)) = 24

  • 3899Solution

    8 × (3 + (9 − 9)) = 24

  • 381010Solution

    8 × (3 + (10 − 10)) = 24

  • 381011Solution

    8 × (3 × (11 − 10)) = 24

  • 381111Solution

    8 × (3 + (11 − 11)) = 24

  • 381112Solution

    8 × (3 × (12 − 11)) = 24

  • 381212Solution

    8 × (3 + (12 − 12)) = 24

  • 381213Solution

    8 × (3 × (13 − 12)) = 24

  • 381313Solution

    8 × (3 + (13 − 13)) = 24

  • 4446Solution

    6 × (4 + (4 − 4)) = 24

  • 4456Solution

    6 × (4 × (5 − 4)) = 24

  • 44812Solution

    12 × ((4 × 4) ÷ 8) = 24

  • 4556Solution

    6 × (4 + (5 − 5)) = 24

  • 4566Solution

    6 × (4 × (6 − 5)) = 24

  • 451012Solution

    12 × ((5 × 4) ÷ 10) = 24

  • 451113Solution

    13 + (11 × (5 − 4)) = 24

  • 451212Solution

    (12 × (5 − 4)) + 12 = 24

  • 4666Solution

    6 × (4 + (6 − 6)) = 24

  • 4667Solution

    6 × (4 × (7 − 6)) = 24

  • 4678Solution

    6 × (4 × (8 − 7)) = 24

  • 4688Solution

    6 × (4 + (8 − 8)) = 24

  • 4689Solution

    6 × (4 × (9 − 8)) = 24

  • 4699Solution

    6 × (4 + (9 − 9)) = 24

  • 46910Solution

    6 × (4 × (10 − 9)) = 24

  • 461010Solution

    6 × (4 + (10 − 10)) = 24

  • 461011Solution

    6 × (4 × (11 − 10)) = 24

  • 461112Solution

    6 × (4 × (12 − 11)) = 24

  • 461212Solution

    6 × (4 + (12 − 12)) = 24

  • 461213Solution

    6 × (4 × (13 − 12)) = 24

  • 48912Solution

    8 × (9 ÷ (12 ÷ 4)) = 24

  • 56810Solution

    8 × (6 ÷ (10 ÷ 5)) = 24

  • 561113Solution

    13 + (11 × (6 − 5)) = 24

  • 561212Solution

    (12 × (6 − 5)) + 12 = 24

  • 66812Solution

    8 × (6 ÷ (12 ÷ 6)) = 24

  • 671113Solution

    13 + (11 × (7 − 6)) = 24

  • 671212Solution

    (12 × (7 − 6)) + 12 = 24

  • 781113Solution

    13 + (11 × (8 − 7)) = 24

  • 781212Solution

    (12 × (8 − 7)) + 12 = 24

  • 891113Solution

    13 + (11 × (9 − 8)) = 24

  • 891212Solution

    (12 × (9 − 8)) + 12 = 24

  • 9101113Solution

    13 + (11 × (10 − 9)) = 24

  • 9101212Solution

    (12 × (10 − 9)) + 12 = 24

  • 10111113Solution

    13 + (11 × (11 − 10)) = 24

  • 10111212Solution

    (12 × (11 − 10)) + 12 = 24

  • 11111213Solution

    13 + (11 × (12 − 11)) = 24

  • 11121212Solution

    (12 × (12 − 11)) + 12 = 24

  • 11121313Solution

    13 + (11 × (13 − 12)) = 24

  • 12121213Solution

    (12 × (13 − 12)) + 12 = 24

3Construct familiar factors

Look for 4 × 6, 3 × 8, or 2 × 12. The factors may come from two sums, a sum and a difference, or two differences. Recognizing that last pattern is a useful skill in its own right.

45811
Reveal solutionHide solution for level 3

(11 − 5) × (8 − 4) = 24

Two differences produce 6 and 4. Neither factor appears as a ready-made pair in the hand.

Browse all 582 problemsHide 582 problems · Level 3

Ordered by the four numbers. Select a hand to reveal one of its easiest solutions. Scroll the list to see every problem in this level.

  • 1118Solution

    8 × ((1 + 1) + 1) = 24

  • 11111Solution

    (11 + 1) × (1 + 1) = 24

  • 11113Solution

    (13 − 1) × (1 + 1) = 24

  • 1127Solution

    (7 + 1) × (2 + 1) = 24

  • 1129Solution

    (9 − 1) × (2 + 1) = 24

  • 11210Solution

    (10 + (1 + 1)) × 2 = 24

  • 1135Solution

    (5 + 1) × (3 + 1) = 24

  • 11310Solution

    (10 − (1 + 1)) × 3 = 24

  • 11311Solution

    (11 + 1) × (3 − 1) = 24

  • 11313Solution

    (13 − 1) × (3 − 1) = 24

  • 1144Solution

    (4 + (1 + 1)) × 4 = 24

  • 1149Solution

    (9 − 1) × (4 − 1) = 24

  • 11412Solution

    12 × (4 − (1 + 1)) = 24

  • 1155Solution

    (5 + 1) × (5 − 1) = 24

  • 1157Solution

    (7 + 5) × (1 + 1) = 24

  • 1158Solution

    8 × (5 − (1 + 1)) = 24

  • 1166Solution

    (6 + 6) × (1 + 1) = 24

  • 1168Solution

    8 × (6 ÷ (1 + 1)) = 24

  • 1228Solution

    8 × ((2 + 2) − 1) = 24

  • 1229Solution

    (9 + (2 + 1)) × 2 = 24

  • 12211Solution

    ((11 + 2) − 1) × 2 = 24

  • 12213Solution

    ((13 + 1) − 2) × 2 = 24

  • 1237Solution

    ((7 + 2) − 1) × 3 = 24

  • 12311Solution

    (11 − 3) × (2 + 1) = 24

  • 12312Solution

    12 × ((3 + 1) − 2) = 24

  • 1245Solution

    ((5 + 2) − 1) × 4 = 24

  • 1247Solution

    (7 + (4 + 1)) × 2 = 24

  • 1249Solution

    ((9 + 4) − 1) × 2 = 24

  • 12410Solution

    (10 − 2) × (4 − 1) = 24

  • 12411Solution

    (11 + 1) × (4 − 2) = 24

  • 12413Solution

    (13 − 1) × (4 − 2) = 24

  • 1256Solution

    ((5 + 1) + 6) × 2 = 24

  • 1259Solution

    (9 − 1) × (5 − 2) = 24

  • 12512Solution

    12 × (5 − (2 + 1)) = 24

  • 12513Solution

    (13 − 5) × (2 + 1) = 24

  • 1267Solution

    ((7 + 6) − 1) × 2 = 24

  • 1269Solution

    6 × ((9 − 1) ÷ 2) = 24

  • 12610Solution

    6 × ((10 ÷ 2) − 1) = 24

  • 12612Solution

    6 × (12 ÷ (2 + 1)) = 24

  • 1278Solution

    8 × ((7 − 1) ÷ 2) = 24

  • 1288Solution

    8 × ((8 ÷ 2) − 1) = 24

  • 1289Solution

    8 × (9 ÷ (2 + 1)) = 24

  • 1333Solution

    (3 + 3) × (3 + 1) = 24

  • 1336Solution

    ((6 + 3) − 1) × 3 = 24

  • 1339Solution

    (9 − 3) × (3 + 1) = 24

  • 13310Solution

    ((10 + 1) − 3) × 3 = 24

  • 13312Solution

    (12 − (3 + 1)) × 3 = 24

  • 1345Solution

    ((5 + 4) − 1) × 3 = 24

  • 1348Solution

    ((8 + 1) − 3) × 4 = 24

  • 13410Solution

    (10 − 4) × (3 + 1) = 24

  • 13411Solution

    ((11 + 1) − 4) × 3 = 24

  • 13413Solution

    (13 − (4 + 1)) × 3 = 24

  • 1357Solution

    (5 + 1) × (7 − 3) = 24

  • 1358Solution

    8 × ((5 + 1) − 3) = 24

  • 1359Solution

    (9 − 3) × (5 − 1) = 24

  • 13511Solution

    (11 + 1) × (5 − 3) = 24

  • 1366Solution

    6 × ((6 + 1) − 3) = 24

  • 1369Solution

    (9 − 1) × (6 − 3) = 24

  • 13611Solution

    6 × ((11 + 1) ÷ 3) = 24

  • 13613Solution

    ((13 + 1) − 6) × 3 = 24

  • 1377Solution

    (7 − 1) × (7 − 3) = 24

  • 1378Solution

    8 × (7 − (3 + 1)) = 24

  • 1379Solution

    (7 + 1) × (9 ÷ 3) = 24

  • 13712Solution

    12 × ((7 − 1) ÷ 3) = 24

  • 1388Solution

    8 × ((8 + 1) ÷ 3) = 24

  • 13810Solution

    8 × ((10 − 1) ÷ 3) = 24

  • 1399Solution

    (9 − 1) × (9 ÷ 3) = 24

  • 1444Solution

    (4 + 4) × (4 − 1) = 24

  • 1449Solution

    ((9 + 1) − 4) × 4 = 24

  • 14411Solution

    (11 − (4 + 1)) × 4 = 24

  • 14412Solution

    (12 − 4) × (4 − 1) = 24

  • 1458Solution

    (5 + 1) × (8 − 4) = 24

  • 14510Solution

    ((10 + 1) − 5) × 4 = 24

  • 14512Solution

    12 × ((5 + 1) − 4) = 24

  • 14513Solution

    (13 − 5) × (4 − 1) = 24

  • 1467Solution

    6 × ((7 + 1) − 4) = 24

  • 1469Solution

    6 × (9 − (4 + 1)) = 24

  • 14611Solution

    ((11 + 1) − 6) × 4 = 24

  • 1477Solution

    (7 + 1) × (7 − 4) = 24

  • 1479Solution

    (9 − 1) × (7 − 4) = 24

  • 1488Solution

    8 × (8 − (4 + 1)) = 24

  • 1489Solution

    8 × (9 ÷ (4 − 1)) = 24

  • 141012Solution

    12 × (10 ÷ (4 + 1)) = 24

  • 141212Solution

    12 × ((12 ÷ 4) − 1) = 24

  • 1559Solution

    (5 + 1) × (9 − 5) = 24

  • 15511Solution

    (11 − 5) × (5 − 1) = 24

  • 15512Solution

    12 × ((5 ÷ 5) + 1) = 24

  • 1568Solution

    6 × ((8 + 1) − 5) = 24

  • 15610Solution

    (5 + 1) × (10 − 6) = 24

  • 1578Solution

    8 × ((7 + 1) − 5) = 24

  • 1579Solution

    (7 − 1) × (9 − 5) = 24

  • 1589Solution

    8 × (9 − (5 + 1)) = 24

  • 15912Solution

    12 × ((9 + 1) ÷ 5) = 24

  • 15913Solution

    (5 + 1) × (13 − 9) = 24

  • 151011Solution

    (11 + 1) × (10 ÷ 5) = 24

  • 151013Solution

    (13 − 1) × (10 ÷ 5) = 24

  • 151112Solution

    12 × ((11 − 1) ÷ 5) = 24

  • 151212Solution

    12 × (12 ÷ (5 + 1)) = 24

  • 1669Solution

    6 × ((9 + 1) − 6) = 24

  • 1679Solution

    (7 + 1) × (9 − 6) = 24

  • 1688Solution

    8 × ((8 + 1) − 6) = 24

  • 16813Solution

    6 × (13 − (8 + 1)) = 24

  • 16912Solution

    6 × ((12 + 1) − 9) = 24

  • 161012Solution

    12 × (10 ÷ (6 − 1)) = 24

  • 161013Solution

    6 × ((13 + 1) − 10) = 24

  • 161112Solution

    12 × ((11 + 1) ÷ 6) = 24

  • 161213Solution

    12 × ((13 − 1) ÷ 6) = 24

  • 17712Solution

    12 × ((7 ÷ 7) + 1) = 24

  • 17811Solution

    (7 + 1) × (11 − 8) = 24

  • 17812Solution

    12 × ((8 + 1) − 7) = 24

  • 17910Solution

    (9 − 1) × (10 − 7) = 24

  • 17911Solution

    (11 + 1) × (9 − 7) = 24

  • 17913Solution

    (13 − 1) × (9 − 7) = 24

  • 171012Solution

    12 × (10 − (7 + 1)) = 24

  • 171013Solution

    (7 + 1) × (13 − 10) = 24

  • 171212Solution

    12 × (12 ÷ (7 − 1)) = 24

  • 171213Solution

    12 × ((13 + 1) ÷ 7) = 24

  • 18810Solution

    8 × ((10 + 1) − 8) = 24

  • 18812Solution

    8 × (12 − (8 + 1)) = 24

  • 18911Solution

    8 × ((11 + 1) − 9) = 24

  • 18913Solution

    8 × (13 − (9 + 1)) = 24

  • 181011Solution

    (11 + 1) × (10 − 8) = 24

  • 181112Solution

    12 × (11 − (8 + 1)) = 24

  • 181113Solution

    8 × ((13 + 1) − 11) = 24

  • 19912Solution

    (9 − 1) × (12 − 9) = 24

  • 191012Solution

    12 × ((10 + 1) − 9) = 24

  • 191013Solution

    (9 − 1) × (13 − 10) = 24

  • 191111Solution

    (11 + 1) × (11 − 9) = 24

  • 191113Solution

    (13 − 1) × (11 − 9) = 24

  • 191212Solution

    12 × (12 − (9 + 1)) = 24

  • 1101012Solution

    12 × ((10 ÷ 10) + 1) = 24

  • 1101112Solution

    12 × ((11 + 1) − 10) = 24

  • 1101213Solution

    12 × (13 − (10 + 1)) = 24

  • 1111112Solution

    12 × ((11 ÷ 11) + 1) = 24

  • 1111113Solution

    (11 + 1) × (13 − 11) = 24

  • 1111212Solution

    12 × ((12 + 1) − 11) = 24

  • 1111313Solution

    (13 − 1) × (13 − 11) = 24

  • 1121212Solution

    12 × ((12 ÷ 12) + 1) = 24

  • 1121213Solution

    12 × ((13 + 1) − 12) = 24

  • 1121313Solution

    12 × ((13 ÷ 13) + 1) = 24

  • 2225Solution

    ((5 × 2) + 2) × 2 = 24

  • 2227Solution

    ((7 × 2) − 2) × 2 = 24

  • 22211Solution

    (11 + (2 ÷ 2)) × 2 = 24

  • 22213Solution

    (13 − (2 ÷ 2)) × 2 = 24

  • 2234Solution

    ((2 + 2) + 4) × 3 = 24

  • 2235Solution

    ((5 × 2) − 2) × 3 = 24

  • 2237Solution

    (7 + (3 + 2)) × 2 = 24

  • 22311Solution

    ((11 + 3) − 2) × 2 = 24

  • 22313Solution

    ((13 + 2) − 3) × 2 = 24

  • 2244Solution

    ((4 × 2) + 4) × 2 = 24

  • 2247Solution

    (7 − (2 ÷ 2)) × 4 = 24

  • 2248Solution

    ((8 × 2) − 4) × 2 = 24

  • 2255Solution

    ((5 + 2) + 5) × 2 = 24

  • 2256Solution

    (6 + 2) × (5 − 2) = 24

  • 2259Solution

    ((9 + 5) − 2) × 2 = 24

  • 22510Solution

    (10 − 2) × (5 − 2) = 24

  • 2266Solution

    6 × ((6 + 2) ÷ 2) = 24

  • 2268Solution

    ((8 + 6) − 2) × 2 = 24

  • 2269Solution

    ((9 × 2) − 6) × 2 = 24

  • 22610Solution

    6 × ((10 − 2) ÷ 2) = 24

  • 2277Solution

    ((7 + 7) − 2) × 2 = 24

  • 2278Solution

    8 × (7 − (2 + 2)) = 24

  • 22710Solution

    (7 + (10 ÷ 2)) × 2 = 24

  • 2288Solution

    (8 + (8 ÷ 2)) × 2 = 24

  • 22810Solution

    ((10 × 2) − 8) × 2 = 24

  • 221011Solution

    ((11 × 2) − 10) × 2 = 24

  • 2333Solution

    ((3 + 2) + 3) × 3 = 24

  • 2335Solution

    ((5 × 3) − 3) × 2 = 24

  • 2336Solution

    ((3 + 3) + 6) × 2 = 24

  • 2339Solution

    ((9 + 2) − 3) × 3 = 24

  • 23310Solution

    ((10 ÷ 2) + 3) × 3 = 24

  • 23311Solution

    (11 + (3 ÷ 3)) × 2 = 24

  • 23313Solution

    (13 − (3 + 2)) × 3 = 24

  • 2345Solution

    ((4 + 3) + 5) × 2 = 24

  • 2347Solution

    ((7 + 2) − 3) × 4 = 24

  • 23410Solution

    ((10 + 2) − 4) × 3 = 24

  • 23411Solution

    ((11 + 4) − 3) × 2 = 24

  • 23413Solution

    ((13 + 3) − 4) × 2 = 24

  • 2355Solution

    ((5 + 5) − 2) × 3 = 24

  • 23510Solution

    ((10 + 5) − 3) × 2 = 24

  • 23511Solution

    ((11 + 2) − 5) × 3 = 24

  • 2367Solution

    ((7 × 2) − 6) × 3 = 24

  • 2369Solution

    ((9 + 6) − 3) × 2 = 24

  • 23611Solution

    6 × ((11 − 3) ÷ 2) = 24

  • 23612Solution

    ((12 + 2) − 6) × 3 = 24

  • 2378Solution

    ((8 + 7) − 3) × 2 = 24

  • 2379Solution

    ((7 × 3) − 9) × 2 = 24

  • 23713Solution

    ((13 + 2) − 7) × 3 = 24

  • 2388Solution

    ((8 × 2) − 8) × 3 = 24

  • 2389Solution

    8 × (9 − (3 × 2)) = 24

  • 2399Solution

    (9 + (9 ÷ 3)) × 2 = 24

  • 231012Solution

    ((10 × 2) − 12) × 3 = 24

  • 2444Solution

    ((4 + 4) + 4) × 2 = 24

  • 2445Solution

    ((5 × 2) − 4) × 4 = 24

  • 2446Solution

    6 × ((4 × 2) − 4) = 24

  • 2448Solution

    ((8 + 2) − 4) × 4 = 24

  • 24411Solution

    (11 + (4 ÷ 4)) × 2 = 24

  • 24413Solution

    (13 − (4 ÷ 4)) × 2 = 24

  • 2457Solution

    (7 + 5) × (4 − 2) = 24

  • 2459Solution

    ((9 + 2) − 5) × 4 = 24

  • 24511Solution

    ((11 + 5) − 4) × 2 = 24

  • 2467Solution

    (6 + 2) × (7 − 4) = 24

  • 24610Solution

    ((10 + 2) − 6) × 4 = 24

  • 2478Solution

    ((7 × 2) − 8) × 4 = 24

  • 2479Solution

    ((9 + 7) − 4) × 2 = 24

  • 2488Solution

    ((8 + 8) − 4) × 2 = 24

  • 2489Solution

    8 × (9 − (4 + 2)) = 24

  • 24812Solution

    ((12 + 2) − 8) × 4 = 24

  • 241011Solution

    (11 − (10 ÷ 2)) × 4 = 24

  • 241212Solution

    12 × (12 ÷ (4 + 2)) = 24

  • 2558Solution

    8 × (2 + (5 ÷ 5)) = 24

  • 25511Solution

    (11 + (5 ÷ 5)) × 2 = 24

  • 25513Solution

    (13 − 5) × (5 − 2) = 24

  • 2566Solution

    6 × ((5 × 2) − 6) = 24

  • 2568Solution

    8 × ((6 + 2) − 5) = 24

  • 25613Solution

    ((13 + 5) − 6) × 2 = 24

  • 2578Solution

    8 × ((5 × 2) − 7) = 24

  • 25810Solution

    8 × (10 − (5 + 2)) = 24

  • 25811Solution

    8 × ((11 − 5) ÷ 2) = 24

  • 25812Solution

    12 × ((5 × 2) − 8) = 24

  • 251010Solution

    (10 + (10 ÷ 5)) × 2 = 24

  • 251212Solution

    12 × (12 − (5 × 2)) = 24

  • 2667Solution

    6 × (7 − (6 ÷ 2)) = 24

  • 2669Solution

    (6 + 2) × (9 − 6) = 24

  • 26611Solution

    (11 + (6 ÷ 6)) × 2 = 24

  • 26613Solution

    (13 − (6 ÷ 6)) × 2 = 24

  • 2678Solution

    8 × ((7 + 2) − 6) = 24

  • 26710Solution

    6 × ((7 × 2) − 10) = 24

  • 26711Solution

    ((11 + 7) − 6) × 2 = 24

  • 26811Solution

    8 × (11 − (6 + 2)) = 24

  • 2699Solution

    ((9 + 9) − 6) × 2 = 24

  • 26910Solution

    (10 − 2) × (9 − 6) = 24

  • 26912Solution

    (6 + 2) × (12 − 9) = 24

  • 261013Solution

    (6 + 2) × (13 − 10) = 24

  • 27711Solution

    (11 + (7 ÷ 7)) × 2 = 24

  • 27713Solution

    (13 − (7 ÷ 7)) × 2 = 24

  • 2788Solution

    8 × ((8 + 2) − 7) = 24

  • 27813Solution

    ((13 + 7) − 8) × 2 = 24

  • 271010Solution

    (10 − 2) × (10 − 7) = 24

  • 271012Solution

    12 × (10 ÷ (7 − 2)) = 24

  • 271112Solution

    12 × (11 − (7 + 2)) = 24

  • 271212Solution

    12 × ((7 × 2) − 12) = 24

  • 2888Solution

    8 × (2 + (8 ÷ 8)) = 24

  • 2889Solution

    8 × ((9 + 2) − 8) = 24

  • 28811Solution

    (11 + (8 ÷ 8)) × 2 = 24

  • 28813Solution

    8 × ((8 × 2) − 13) = 24

  • 28910Solution

    8 × ((10 + 2) − 9) = 24

  • 28911Solution

    ((11 + 9) − 8) × 2 = 24

  • 28913Solution

    ((13 + 8) − 9) × 2 = 24

  • 281010Solution

    ((10 + 10) − 8) × 2 = 24

  • 281011Solution

    8 × ((11 + 2) − 10) = 24

  • 281111Solution

    8 × (2 + (11 ÷ 11)) = 24

  • 281112Solution

    8 × ((12 + 2) − 11) = 24

  • 281212Solution

    12 × (12 − (8 + 2)) = 24

  • 281213Solution

    8 × ((13 + 2) − 12) = 24

  • 281313Solution

    8 × (2 + (13 ÷ 13)) = 24

  • 29911Solution

    (11 + (9 ÷ 9)) × 2 = 24

  • 29913Solution

    (13 − (9 ÷ 9)) × 2 = 24

  • 291011Solution

    ((11 + 10) − 9) × 2 = 24

  • 291013Solution

    ((13 + 9) − 10) × 2 = 24

  • 291213Solution

    12 × (13 − (9 + 2)) = 24

  • 2101011Solution

    (11 + (10 ÷ 10)) × 2 = 24

  • 2101013Solution

    (10 − 2) × (13 − 10) = 24

  • 2101111Solution

    ((11 + 11) − 10) × 2 = 24

  • 2101113Solution

    ((13 + 10) − 11) × 2 = 24

  • 2111111Solution

    (11 + (11 ÷ 11)) × 2 = 24

  • 2111113Solution

    (13 − (11 ÷ 11)) × 2 = 24

  • 2111213Solution

    ((13 + 11) − 12) × 2 = 24

  • 2111313Solution

    (11 + (13 ÷ 13)) × 2 = 24

  • 2131313Solution

    (13 − (13 ÷ 13)) × 2 = 24

  • 3334Solution

    ((3 × 3) − 3) × 4 = 24

  • 3336Solution

    6 × (3 + (3 ÷ 3)) = 24

  • 3337Solution

    (3 + 3) × (7 − 3) = 24

  • 3339Solution

    (9 − (3 ÷ 3)) × 3 = 24

  • 33312Solution

    12 × ((3 + 3) ÷ 3) = 24

  • 3344Solution

    ((4 × 3) − 4) × 3 = 24

  • 3347Solution

    ((7 + 4) − 3) × 3 = 24

  • 3349Solution

    ((9 + 3) − 4) × 3 = 24

  • 33412Solution

    12 × ((3 + 3) − 4) = 24

  • 3356Solution

    6 × ((3 × 3) − 5) = 24

  • 3357Solution

    ((5 × 3) − 7) × 3 = 24

  • 3359Solution

    (3 + 3) × (9 − 5) = 24

  • 33510Solution

    ((10 + 3) − 5) × 3 = 24

  • 3366Solution

    (6 + (6 ÷ 3)) × 3 = 24

  • 3368Solution

    8 × ((3 × 3) − 6) = 24

  • 3369Solution

    6 × ((9 + 3) ÷ 3) = 24

  • 33610Solution

    ((6 × 3) − 10) × 3 = 24

  • 33611Solution

    ((11 + 3) − 6) × 3 = 24

  • 33613Solution

    6 × (13 − (3 × 3)) = 24

  • 3379Solution

    (9 − 3) × (7 − 3) = 24

  • 33712Solution

    ((12 + 3) − 7) × 3 = 24

  • 33713Solution

    ((7 × 3) − 13) × 3 = 24

  • 3389Solution

    8 × (9 − (3 + 3)) = 24

  • 33812Solution

    8 × (12 − (3 × 3)) = 24

  • 33813Solution

    ((13 + 3) − 8) × 3 = 24

  • 33911Solution

    (11 − (9 ÷ 3)) × 3 = 24

  • 33912Solution

    12 × ((9 − 3) ÷ 3) = 24

  • 33913Solution

    (3 + 3) × (13 − 9) = 24

  • 331112Solution

    12 × (11 − (3 × 3)) = 24

  • 3445Solution

    ((5 + 4) − 3) × 4 = 24

  • 3447Solution

    ((7 + 3) − 4) × 4 = 24

  • 3449Solution

    (4 + 4) × (9 ÷ 3) = 24

  • 34412Solution

    ((12 ÷ 4) + 3) × 4 = 24

  • 3456Solution

    6 × ((5 + 3) − 4) = 24

  • 3459Solution

    ((5 × 3) − 9) × 4 = 24

  • 34513Solution

    ((13 + 5) ÷ 3) × 4 = 24

  • 3466Solution

    ((6 + 6) − 4) × 3 = 24

  • 3469Solution

    ((9 + 3) − 6) × 4 = 24

  • 34610Solution

    ((10 + 4) − 6) × 3 = 24

  • 34613Solution

    6 × ((13 + 3) ÷ 4) = 24

  • 34711Solution

    ((11 + 4) − 7) × 3 = 24

  • 34811Solution

    ((11 + 3) − 8) × 4 = 24

  • 34812Solution

    ((12 + 4) − 8) × 3 = 24

  • 34813Solution

    8 × ((13 − 4) ÷ 3) = 24

  • 3499Solution

    ((9 + 9) ÷ 3) × 4 = 24

  • 34912Solution

    ((12 + 3) − 9) × 4 = 24

  • 34913Solution

    ((13 + 4) − 9) × 3 = 24

  • 3556Solution

    6 × (3 + (5 ÷ 5)) = 24

  • 3557Solution

    (7 + 5) × (5 − 3) = 24

  • 3559Solution

    (9 − (5 ÷ 5)) × 3 = 24

  • 35512Solution

    12 × (3 − (5 ÷ 5)) = 24

  • 3566Solution

    6 × ((6 + 3) − 5) = 24

  • 3567Solution

    ((7 + 6) − 5) × 3 = 24

  • 3569Solution

    ((9 + 5) − 6) × 3 = 24

  • 35611Solution

    6 × ((5 × 3) − 11) = 24

  • 35612Solution

    12 × ((5 + 3) − 6) = 24

  • 35613Solution

    (13 − 5) × (6 − 3) = 24

  • 35710Solution

    ((10 + 5) − 7) × 3 = 24

  • 35711Solution

    (11 − 5) × (7 − 3) = 24

  • 35712Solution

    12 × ((7 + 3) ÷ 5) = 24

  • 35811Solution

    ((11 + 5) − 8) × 3 = 24

  • 35812Solution

    8 × ((5 × 3) − 12) = 24

  • 3599Solution

    (9 − 3) × (9 − 5) = 24

  • 35910Solution

    (9 + 3) × (10 ÷ 5) = 24

  • 35912Solution

    ((12 + 5) − 9) × 3 = 24

  • 351010Solution

    (10 − (10 ÷ 5)) × 3 = 24

  • 351013Solution

    ((13 + 5) − 10) × 3 = 24

  • 351112Solution

    12 × ((11 − 5) ÷ 3) = 24

  • 351213Solution

    12 × ((5 × 3) − 13) = 24

  • 3666Solution

    6 × ((6 + 6) ÷ 3) = 24

  • 3667Solution

    6 × ((7 + 3) − 6) = 24

  • 36612Solution

    (6 + (12 ÷ 6)) × 3 = 24

  • 36613Solution

    6 × (13 − (6 + 3)) = 24

  • 3677Solution

    ((7 + 7) − 6) × 3 = 24

  • 3679Solution

    ((9 + 6) − 7) × 3 = 24

  • 36712Solution

    12 × ((6 + 3) − 7) = 24

  • 3688Solution

    6 × (3 + (8 ÷ 8)) = 24

  • 3689Solution

    6 × ((9 + 3) − 8) = 24

  • 36810Solution

    ((10 + 6) − 8) × 3 = 24

  • 36812Solution

    8 × (12 − (6 + 3)) = 24

  • 3699Solution

    6 × (3 + (9 ÷ 9)) = 24

  • 36910Solution

    6 × ((10 + 3) − 9) = 24

  • 36911Solution

    ((11 + 6) − 9) × 3 = 24

  • 361010Solution

    6 × (3 + (10 ÷ 10)) = 24

  • 361012Solution

    ((12 + 6) − 10) × 3 = 24

  • 361111Solution

    6 × (3 + (11 ÷ 11)) = 24

  • 361112Solution

    6 × ((12 + 3) − 11) = 24

  • 361113Solution

    ((13 + 6) − 11) × 3 = 24

  • 361212Solution

    6 × (3 + (12 ÷ 12)) = 24

  • 361213Solution

    6 × ((13 + 3) − 12) = 24

  • 361313Solution

    6 × (3 + (13 ÷ 13)) = 24

  • 3779Solution

    (9 − (7 ÷ 7)) × 3 = 24

  • 37712Solution

    12 × (3 − (7 ÷ 7)) = 24

  • 3789Solution

    ((9 + 7) − 8) × 3 = 24

  • 37813Solution

    8 × (13 − (7 + 3)) = 24

  • 3799Solution

    (9 + 3) × (9 − 7) = 24

  • 37910Solution

    ((10 + 7) − 9) × 3 = 24

  • 371011Solution

    ((11 + 7) − 10) × 3 = 24

  • 371111Solution

    (7 + (11 ÷ 11)) × 3 = 24

  • 371112Solution

    ((12 + 7) − 11) × 3 = 24

  • 371212Solution

    12 × (12 − (7 + 3)) = 24

  • 371213Solution

    ((13 + 7) − 12) × 3 = 24

  • 371313Solution

    (7 + (13 ÷ 13)) × 3 = 24

  • 38812Solution

    12 × (3 − (8 ÷ 8)) = 24

  • 38912Solution

    12 × ((8 + 3) − 9) = 24

  • 381012Solution

    12 × (10 ÷ (8 − 3)) = 24

  • 39910Solution

    ((9 + 9) − 10) × 3 = 24

  • 39911Solution

    (9 + 3) × (11 − 9) = 24

  • 39912Solution

    12 × (3 − (9 ÷ 9)) = 24

  • 39913Solution

    (9 − 3) × (13 − 9) = 24

  • 391010Solution

    (9 − (10 ÷ 10)) × 3 = 24

  • 391011Solution

    ((10 + 9) − 11) × 3 = 24

  • 391012Solution

    12 × ((9 + 3) − 10) = 24

  • 391111Solution

    (9 − (11 ÷ 11)) × 3 = 24

  • 391112Solution

    ((11 + 9) − 12) × 3 = 24

  • 391113Solution

    (9 + 3) × (13 − 11) = 24

  • 391212Solution

    12 × (12 ÷ (9 − 3)) = 24

  • 391213Solution

    ((12 + 9) − 13) × 3 = 24

  • 391313Solution

    (9 − (13 ÷ 13)) × 3 = 24

  • 3101012Solution

    ((10 + 10) − 12) × 3 = 24

  • 3101112Solution

    12 × ((10 + 3) − 11) = 24

  • 3101113Solution

    ((11 + 10) − 13) × 3 = 24

  • 3111112Solution

    12 × (3 − (11 ÷ 11)) = 24

  • 3111212Solution

    12 × ((11 + 3) − 12) = 24

  • 3121212Solution

    12 × (3 − (12 ÷ 12)) = 24

  • 3121213Solution

    12 × ((12 + 3) − 13) = 24

  • 3121313Solution

    12 × (3 − (13 ÷ 13)) = 24

  • 4445Solution

    (5 + (4 ÷ 4)) × 4 = 24

  • 4447Solution

    (4 + 4) × (7 − 4) = 24

  • 4448Solution

    (4 + (8 ÷ 4)) × 4 = 24

  • 44410Solution

    ((4 × 4) − 10) × 4 = 24

  • 4455Solution

    ((5 + 5) − 4) × 4 = 24

  • 4457Solution

    ((7 + 4) − 5) × 4 = 24

  • 4458Solution

    8 × ((4 + 4) − 5) = 24

  • 44510Solution

    (4 + (10 ÷ 5)) × 4 = 24

  • 4468Solution

    ((8 + 4) − 6) × 4 = 24

  • 4469Solution

    (4 + 4) × (9 − 6) = 24

  • 44612Solution

    12 × ((4 + 4) − 6) = 24

  • 44710Solution

    (4 + 4) × (10 − 7) = 24

  • 44712Solution

    (12 − 4) × (7 − 4) = 24

  • 44810Solution

    ((10 + 4) − 8) × 4 = 24

  • 44811Solution

    8 × (11 − (4 + 4)) = 24

  • 44813Solution

    8 × ((4 × 4) − 13) = 24

  • 44911Solution

    ((11 + 4) − 9) × 4 = 24

  • 44912Solution

    (4 + 4) × (12 − 9) = 24

  • 441012Solution

    ((12 + 4) − 10) × 4 = 24

  • 441013Solution

    (4 + 4) × (13 − 10) = 24

  • 4555Solution

    (5 + (5 ÷ 5)) × 4 = 24

  • 4557Solution

    (7 − (5 ÷ 5)) × 4 = 24

  • 4558Solution

    8 × (4 − (5 ÷ 5)) = 24

  • 4567Solution

    ((7 + 5) − 6) × 4 = 24

  • 4568Solution

    8 × ((5 + 4) − 6) = 24

  • 45611Solution

    6 × ((11 + 5) ÷ 4) = 24

  • 45612Solution

    12 × ((6 + 4) ÷ 5) = 24

  • 45613Solution

    6 × (13 − (5 + 4)) = 24

  • 4577Solution

    (5 + (7 ÷ 7)) × 4 = 24

  • 45712Solution

    12 × ((5 + 4) − 7) = 24

  • 45713Solution

    (13 − 5) × (7 − 4) = 24

  • 4588Solution

    (5 + (8 ÷ 8)) × 4 = 24

  • 4589Solution

    ((9 + 5) − 8) × 4 = 24

  • 45810Solution

    (8 + 4) × (10 ÷ 5) = 24

  • 45811Solution

    (11 − 5) × (8 − 4) = 24

  • 45812Solution

    8 × (12 − (5 + 4)) = 24

  • 4599Solution

    (5 + (9 ÷ 9)) × 4 = 24

  • 45910Solution

    ((10 + 5) − 9) × 4 = 24

  • 451010Solution

    (5 + (10 ÷ 10)) × 4 = 24

  • 451011Solution

    ((11 + 5) − 10) × 4 = 24

  • 451111Solution

    (5 + (11 ÷ 11)) × 4 = 24

  • 451313Solution

    (5 + (13 ÷ 13)) × 4 = 24

  • 46610Solution

    6 × ((10 + 6) ÷ 4) = 24

  • 4679Solution

    6 × ((9 + 7) ÷ 4) = 24

  • 46712Solution

    6 × (12 ÷ (7 − 4)) = 24

  • 46812Solution

    12 × ((6 + 4) − 8) = 24

  • 46813Solution

    8 × (13 − (6 + 4)) = 24

  • 46912Solution

    (12 − 4) × (9 − 6) = 24

  • 4777Solution

    (7 − (7 ÷ 7)) × 4 = 24

  • 4778Solution

    ((7 + 7) − 8) × 4 = 24

  • 4788Solution

    8 × ((7 + 4) − 8) = 24

  • 4789Solution

    ((8 + 7) − 9) × 4 = 24

  • 4799Solution

    (7 − (9 ÷ 9)) × 4 = 24

  • 47910Solution

    ((9 + 7) − 10) × 4 = 24

  • 471010Solution

    (7 − (10 ÷ 10)) × 4 = 24

  • 471012Solution

    (12 − 4) × (10 − 7) = 24

  • 471111Solution

    (7 − (11 ÷ 11)) × 4 = 24

  • 471112Solution

    ((11 + 7) − 12) × 4 = 24

  • 471212Solution

    (7 − (12 ÷ 12)) × 4 = 24

  • 471213Solution

    ((12 + 7) − 13) × 4 = 24

  • 471313Solution

    (7 − (13 ÷ 13)) × 4 = 24

  • 4888Solution

    8 × (4 − (8 ÷ 8)) = 24

  • 4889Solution

    8 × ((8 + 4) − 9) = 24

  • 48810Solution

    ((8 + 8) − 10) × 4 = 24

  • 4899Solution

    8 × (4 − (9 ÷ 9)) = 24

  • 48910Solution

    8 × ((9 + 4) − 10) = 24

  • 481011Solution

    8 × ((10 + 4) − 11) = 24

  • 481012Solution

    ((10 + 8) − 12) × 4 = 24

  • 481111Solution

    8 × (4 − (11 ÷ 11)) = 24

  • 481112Solution

    8 × ((11 + 4) − 12) = 24

  • 481113Solution

    ((11 + 8) − 13) × 4 = 24

  • 481212Solution

    12 × ((12 + 4) ÷ 8) = 24

  • 481213Solution

    8 × ((12 + 4) − 13) = 24

  • 481313Solution

    8 × (4 − (13 ÷ 13)) = 24

  • 49912Solution

    ((9 + 9) − 12) × 4 = 24

  • 491012Solution

    12 × (10 ÷ (9 − 4)) = 24

  • 491013Solution

    ((10 + 9) − 13) × 4 = 24

  • 491112Solution

    12 × ((9 + 4) − 11) = 24

  • 491212Solution

    (12 − 4) × (12 − 9) = 24

  • 4101212Solution

    12 × ((10 + 4) − 12) = 24

  • 4101213Solution

    (12 − 4) × (13 − 10) = 24

  • 4111213Solution

    12 × ((11 + 4) − 13) = 24

  • 5556Solution

    6 × (5 − (5 ÷ 5)) = 24

  • 55512Solution

    12 × ((5 + 5) ÷ 5) = 24

  • 5566Solution

    6 × ((5 + 5) − 6) = 24

  • 5578Solution

    8 × ((5 + 5) − 7) = 24

  • 55710Solution

    (7 + 5) × (10 ÷ 5) = 24

  • 55810Solution

    8 × ((10 + 5) ÷ 5) = 24

  • 55812Solution

    12 × ((5 + 5) − 8) = 24

  • 55813Solution

    8 × (13 − (5 + 5)) = 24

  • 55911Solution

    (11 − 5) × (9 − 5) = 24

  • 5666Solution

    6 × (5 − (6 ÷ 6)) = 24

  • 56610Solution

    (6 + 6) × (10 ÷ 5) = 24

  • 5677Solution

    6 × (5 − (7 ÷ 7)) = 24

  • 5678Solution

    6 × ((7 + 5) − 8) = 24

  • 56712Solution

    12 × ((7 + 5) ÷ 6) = 24

  • 56713Solution

    6 × ((13 + 7) ÷ 5) = 24

  • 5688Solution

    8 × ((6 + 5) − 8) = 24

  • 5689Solution

    6 × ((8 + 5) − 9) = 24

  • 56812Solution

    6 × ((12 + 8) ÷ 5) = 24

  • 56813Solution

    8 × ((13 + 5) ÷ 6) = 24

  • 5699Solution

    6 × (5 − (9 ÷ 9)) = 24

  • 56910Solution

    6 × ((9 + 5) − 10) = 24

  • 56911Solution

    6 × ((11 + 9) ÷ 5) = 24

  • 56912Solution

    12 × ((6 + 5) − 9) = 24

  • 56913Solution

    (13 − 5) × (9 − 6) = 24

  • 561010Solution

    6 × ((10 + 10) ÷ 5) = 24

  • 561011Solution

    6 × ((10 + 5) − 11) = 24

  • 561111Solution

    6 × (5 − (11 ÷ 11)) = 24

  • 561313Solution

    6 × (5 − (13 ÷ 13)) = 24

  • 5779Solution

    (7 + 5) × (9 − 7) = 24

  • 5788Solution

    8 × ((8 + 7) ÷ 5) = 24

  • 5789Solution

    8 × ((7 + 5) − 9) = 24

  • 57810Solution

    (7 + 5) × (10 − 8) = 24

  • 57911Solution

    (7 + 5) × (11 − 9) = 24

  • 57912Solution

    12 × ((9 + 5) ÷ 7) = 24

  • 571013Solution

    (13 − 5) × (10 − 7) = 24

  • 571113Solution

    (7 + 5) × (13 − 11) = 24

  • 5889Solution

    8 × (9 ÷ (8 − 5)) = 24

  • 58810Solution

    8 × ((8 + 5) − 10) = 24

  • 58911Solution

    8 × ((9 + 5) − 11) = 24

  • 581012Solution

    8 × ((10 + 5) − 12) = 24

  • 581112Solution

    12 × ((8 + 5) − 11) = 24

  • 581113Solution

    8 × ((11 + 5) − 13) = 24

  • 591113Solution

    (11 − 5) × (13 − 9) = 24

  • 591212Solution

    12 × ((9 + 5) − 12) = 24

  • 591213Solution

    (13 − 5) × (12 − 9) = 24

  • 5101012Solution

    12 × (10 ÷ (10 − 5)) = 24

  • 5101213Solution

    12 × ((10 + 5) − 13) = 24

  • 5101313Solution

    (13 − 5) × (13 − 10) = 24

  • 5111212Solution

    12 × (12 ÷ (11 − 5)) = 24

  • 6668Solution

    6 × ((6 + 6) − 8) = 24

  • 66612Solution

    12 × ((6 + 6) ÷ 6) = 24

  • 6679Solution

    6 × ((7 + 6) − 9) = 24

  • 6688Solution

    8 × (6 ÷ (8 − 6)) = 24

  • 6689Solution

    8 × ((6 + 6) − 9) = 24

  • 66810Solution

    6 × ((8 + 6) − 10) = 24

  • 66911Solution

    6 × ((9 + 6) − 11) = 24

  • 66912Solution

    6 × (12 ÷ (9 − 6)) = 24

  • 661012Solution

    6 × ((10 + 6) − 12) = 24

  • 67710Solution

    6 × ((7 + 7) − 10) = 24

  • 6789Solution

    8 × (6 ÷ (9 − 7)) = 24

  • 67810Solution

    8 × ((7 + 6) − 10) = 24

  • 67811Solution

    6 × ((8 + 7) − 11) = 24

  • 67812Solution

    12 × ((8 + 6) ÷ 7) = 24

  • 67912Solution

    6 × ((9 + 7) − 12) = 24

  • 671012Solution

    6 × (12 ÷ (10 − 7)) = 24

  • 68810Solution

    8 × ((10 + 8) ÷ 6) = 24

  • 68811Solution

    8 × ((8 + 6) − 11) = 24

  • 68812Solution

    6 × ((8 + 8) − 12) = 24

  • 6899Solution

    8 × ((9 + 9) ÷ 6) = 24

  • 68911Solution

    8 × (6 ÷ (11 − 9)) = 24

  • 68912Solution

    8 × ((9 + 6) − 12) = 24

  • 681013Solution

    8 × ((10 + 6) − 13) = 24

  • 681112Solution

    6 × (12 ÷ (11 − 8)) = 24

  • 681113Solution

    8 × (6 ÷ (13 − 11)) = 24

  • 681212Solution

    12 × ((8 + 6) − 12) = 24

  • 691212Solution

    12 × ((12 + 6) ÷ 9) = 24

  • 691213Solution

    12 × ((9 + 6) − 13) = 24

  • 6101112Solution

    12 × (10 ÷ (11 − 6)) = 24

  • 6101213Solution

    6 × (12 ÷ (13 − 10)) = 24

  • 6121212Solution

    12 × (12 ÷ (12 − 6)) = 24

  • 77712Solution

    12 × ((7 + 7) ÷ 7) = 24

  • 77811Solution

    8 × ((7 + 7) − 11) = 24

  • 78812Solution

    8 × ((8 + 7) − 12) = 24

  • 78813Solution

    8 × ((13 + 8) ÷ 7) = 24

  • 78910Solution

    8 × (9 ÷ (10 − 7)) = 24

  • 78912Solution

    8 × ((12 + 9) ÷ 7) = 24

  • 78913Solution

    8 × ((9 + 7) − 13) = 24

  • 781011Solution

    8 × ((11 + 10) ÷ 7) = 24

  • 791112Solution

    12 × ((11 + 7) ÷ 9) = 24

  • 7101212Solution

    12 × (10 ÷ (12 − 7)) = 24

  • 7101213Solution

    12 × ((13 + 7) ÷ 10) = 24

  • 7121213Solution

    12 × (12 ÷ (13 − 7)) = 24

  • 88812Solution

    12 × ((8 + 8) ÷ 8) = 24

  • 88813Solution

    8 × ((8 + 8) − 13) = 24

  • 88911Solution

    8 × (9 ÷ (11 − 8)) = 24

  • 89912Solution

    8 × (9 ÷ (12 − 9)) = 24

  • 891012Solution

    12 × ((10 + 8) ÷ 9) = 24

  • 8101212Solution

    12 × ((12 + 8) ÷ 10) = 24

  • 8101213Solution

    12 × (10 ÷ (13 − 8)) = 24

  • 99912Solution

    12 × ((9 + 9) ÷ 9) = 24

  • 9111213Solution

    12 × ((13 + 9) ÷ 11) = 24

  • 10101012Solution

    12 × ((10 + 10) ÷ 10) = 24

  • 11111112Solution

    12 × ((11 + 11) ÷ 11) = 24

4Nearby totals and small corrections

If 24 is awkward to build directly, aim nearby. Think 25 − 1, 21 + 3, or 28 − 4. One part of the hand gets close; the remaining part supplies the correction.

5555
Reveal solutionHide solution for level 4

5 × 5 − 5 ÷ 5 = 24

Two 5s make 25. The other two make the 1 we need to subtract.

Browse all 243 problemsHide 243 problems · Level 4

Ordered by the four numbers. Select a hand to reveal one of its easiest solutions. Scroll the list to see every problem in this level.

  • 11410Solution

    (10 × (1 + 1)) + 4 = 24

  • 1169Solution

    (9 × (1 + 1)) + 6 = 24

  • 11612Solution

    (6 × (1 + 1)) + 12 = 24

  • 11710Solution

    (7 × (1 + 1)) + 10 = 24

  • 1188Solution

    (8 × (1 + 1)) + 8 = 24

  • 12313Solution

    ((13 × 2) − 3) + 1 = 24

  • 1255Solution

    ((5 × 5) − 2) + 1 = 24

  • 12510Solution

    ((10 × 2) − 1) + 5 = 24

  • 12611Solution

    ((6 × 2) + 1) + 11 = 24

  • 12613Solution

    13 + ((6 × 2) − 1) = 24

  • 1279Solution

    ((7 × 2) + 1) + 9 = 24

  • 12710Solution

    (7 × (2 × 1)) + 10 = 24

  • 12711Solution

    ((7 × 2) − 1) + 11 = 24

  • 12712Solution

    ((7 − 1) × 2) + 12 = 24

  • 12810Solution

    ((8 − 1) × 2) + 10 = 24

  • 1337Solution

    (7 × (3 × 1)) + 3 = 24

  • 1338Solution

    ((8 − 1) × 3) + 3 = 24

  • 1347Solution

    ((7 × 3) − 1) + 4 = 24

  • 1356Solution

    ((5 + 1) × 3) + 6 = 24

  • 13510Solution

    ((5 × 3) − 1) + 10 = 24

  • 13610Solution

    ((10 × 3) × 1) − 6 = 24

  • 13710Solution

    (7 × (3 − 1)) + 10 = 24

  • 1445Solution

    (5 × (4 × 1)) + 4 = 24

  • 1446Solution

    ((6 − 1) × 4) + 4 = 24

  • 1447Solution

    ((4 × 4) + 1) + 7 = 24

  • 1448Solution

    ((4 × 1) × 4) + 8 = 24

  • 1455Solution

    ((5 × 4) − 1) + 5 = 24

  • 1457Solution

    ((7 × 4) − 5) + 1 = 24

  • 1459Solution

    (5 × (4 − 1)) + 9 = 24

  • 1466Solution

    (6 × (4 − 1)) + 6 = 24

  • 14610Solution

    (10 × (4 − 1)) − 6 = 24

  • 1556Solution

    ((5 + 1) × 5) − 6 = 24

  • 15510Solution

    ((10 − 5) × 5) − 1 = 24

  • 1566Solution

    (6 × (5 × 1)) − 6 = 24

  • 1567Solution

    ((7 − 1) × 5) − 6 = 24

  • 15611Solution

    ((11 − 6) × 5) − 1 = 24

  • 15813Solution

    ((13 − 8) × 5) − 1 = 24

  • 1666Solution

    (6 × (6 − 1)) − 6 = 24

  • 2229Solution

    ((9 + 2) × 2) + 2 = 24

  • 22210Solution

    ((10 × 2) + 2) + 2 = 24

  • 22310Solution

    ((10 + 3) × 2) − 2 = 24

  • 2249Solution

    ((9 × 2) + 2) + 4 = 24

  • 22411Solution

    ((11 × 2) − 2) + 4 = 24

  • 22412Solution

    ((4 + 2) × 2) + 12 = 24

  • 22413Solution

    ((13 × 2) − 4) + 2 = 24

  • 2257Solution

    (7 × 2) + (5 × 2) = 24

  • 2258Solution

    ((8 + 5) × 2) − 2 = 24

  • 22512Solution

    (12 + (5 × 2)) + 2 = 24

  • 2267Solution

    ((7 + 2) × 2) + 6 = 24

  • 22611Solution

    ((11 − 2) × 2) + 6 = 24

  • 22613Solution

    ((13 + 2) × 2) − 6 = 24

  • 22712Solution

    ((7 × 2) − 2) + 12 = 24

  • 2289Solution

    ((9 × 2) − 2) + 8 = 24

  • 22910Solution

    ((9 − 2) × 2) + 10 = 24

  • 22912Solution

    (9 × 2) + (12 ÷ 2) = 24

  • 221013Solution

    13 + (10 + (2 ÷ 2)) = 24

  • 221112Solution

    (11 + (2 ÷ 2)) + 12 = 24

  • 221213Solution

    13 + (12 − (2 ÷ 2)) = 24

  • 2357Solution

    ((5 × 3) + 2) + 7 = 24

  • 2358Solution

    ((8 × 2) + 3) + 5 = 24

  • 23513Solution

    (13 + (3 × 2)) + 5 = 24

  • 2368Solution

    ((6 × 3) − 2) + 8 = 24

  • 2377Solution

    ((7 × 2) + 3) + 7 = 24

  • 23710Solution

    ((10 × 2) − 3) + 7 = 24

  • 23810Solution

    (10 + (3 × 2)) + 8 = 24

  • 23912Solution

    ((9 − 3) × 2) + 12 = 24

  • 231010Solution

    ((10 − 3) × 2) + 10 = 24

  • 2449Solution

    ((9 − 2) × 4) − 4 = 24

  • 24410Solution

    ((4 × 4) − 2) + 10 = 24

  • 2455Solution

    ((5 + 5) × 2) + 4 = 24

  • 2456Solution

    ((5 + 4) × 2) + 6 = 24

  • 24510Solution

    ((5 × 2) + 10) + 4 = 24

  • 24611Solution

    ((11 × 2) − 4) + 6 = 24

  • 24613Solution

    ((13 − 4) × 2) + 6 = 24

  • 2477Solution

    ((7 + 7) × 2) − 4 = 24

  • 24813Solution

    (13 × (8 ÷ 4)) − 2 = 24

  • 24910Solution

    ((9 × 2) − 4) + 10 = 24

  • 2557Solution

    ((7 × 2) + 5) + 5 = 24

  • 2559Solution

    ((5 × 2) + 5) + 9 = 24

  • 2569Solution

    (5 × (6 ÷ 2)) + 9 = 24

  • 2577Solution

    ((5 × 2) + 7) + 7 = 24

  • 25711Solution

    ((11 × 2) − 5) + 7 = 24

  • 25713Solution

    (13 × (7 − 5)) − 2 = 24

  • 25910Solution

    ((10 × 2) − 5) + 9 = 24

  • 25911Solution

    ((9 × 2) − 5) + 11 = 24

  • 251012Solution

    ((12 − 5) × 2) + 10 = 24

  • 251013Solution

    (13 × (10 ÷ 5)) − 2 = 24

  • 251112Solution

    ((11 − 5) × 2) + 12 = 24

  • 251213Solution

    (13 + (12 ÷ 2)) + 5 = 24

  • 2666Solution

    ((6 × 2) + 6) + 6 = 24

  • 26813Solution

    (13 + (6 ÷ 2)) + 8 = 24

  • 261011Solution

    (10 + (6 ÷ 2)) + 11 = 24

  • 261112Solution

    (11 × (12 ÷ 6)) + 2 = 24

  • 261212Solution

    ((12 − 6) × 2) + 12 = 24

  • 261213Solution

    (13 × (12 ÷ 6)) − 2 = 24

  • 27911Solution

    ((11 × 2) − 7) + 9 = 24

  • 27913Solution

    13 + ((9 × 2) − 7) = 24

  • 271011Solution

    ((10 × 2) − 7) + 11 = 24

  • 271213Solution

    ((13 − 7) × 2) + 12 = 24

  • 281012Solution

    ((10 × 2) − 8) + 12 = 24

  • 281013Solution

    (13 × (10 − 8)) − 2 = 24

  • 291010Solution

    (10 + (10 ÷ 2)) + 9 = 24

  • 291111Solution

    ((11 × 2) − 9) + 11 = 24

  • 291113Solution

    (13 × (11 − 9)) − 2 = 24

  • 291313Solution

    13 + ((13 + 9) ÷ 2) = 24

  • 2101213Solution

    13 + ((12 + 10) ÷ 2) = 24

  • 3333Solution

    ((3 × 3) × 3) − 3 = 24

  • 3335Solution

    (5 × 3) + (3 × 3) = 24

  • 33310Solution

    ((10 − 3) × 3) + 3 = 24

  • 33411Solution

    (11 + (3 × 3)) + 4 = 24

  • 33413Solution

    ((13 − 4) × 3) − 3 = 24

  • 3355Solution

    (5 × 5) − (3 ÷ 3) = 24

  • 33512Solution

    ((12 − 5) × 3) + 3 = 24

  • 3367Solution

    ((7 × 3) − 3) + 6 = 24

  • 3378Solution

    ((3 × 3) + 7) + 8 = 24

  • 33910Solution

    (10 × 3) − (9 − 3) = 24

  • 331013Solution

    13 + (10 + (3 ÷ 3)) = 24

  • 331213Solution

    13 + (12 − (3 ÷ 3)) = 24

  • 3444Solution

    ((4 + 3) × 4) − 4 = 24

  • 34410Solution

    ((10 − 3) × 4) − 4 = 24

  • 34411Solution

    ((4 × 4) − 3) + 11 = 24

  • 3455Solution

    ((5 × 3) + 4) + 5 = 24

  • 34511Solution

    ((11 × 3) − 4) − 5 = 24

  • 3477Solution

    ((7 × 3) − 4) + 7 = 24

  • 3478Solution

    ((7 − 3) × 4) + 8 = 24

  • 34712Solution

    (7 × (12 ÷ 4)) + 3 = 24

  • 341010Solution

    (10 × 3) − (10 − 4) = 24

  • 3578Solution

    ((7 × 3) − 5) + 8 = 24

  • 3589Solution

    (9 × (8 − 5)) − 3 = 24

  • 351011Solution

    (10 × 3) − (11 − 5) = 24

  • 36610Solution

    (10 × (6 − 3)) − 6 = 24

  • 36710Solution

    (7 × (6 ÷ 3)) + 10 = 24

  • 36713Solution

    ((13 − 7) × 3) + 6 = 24

  • 36913Solution

    (13 + (6 ÷ 3)) + 9 = 24

  • 37710Solution

    (7 × (10 − 7)) + 3 = 24

  • 37811Solution

    (7 × (11 − 8)) + 3 = 24

  • 37912Solution

    ((12 − 7) × 3) + 9 = 24

  • 371013Solution

    (7 × (13 − 10)) + 3 = 24

  • 38911Solution

    (9 × (11 − 8)) − 3 = 24

  • 38913Solution

    ((13 − 8) × 3) + 9 = 24

  • 391013Solution

    (9 × (13 − 10)) − 3 = 24

  • 4444Solution

    ((4 × 4) + 4) + 4 = 24

  • 4449Solution

    ((9 − 4) × 4) + 4 = 24

  • 44411Solution

    ((11 − 4) × 4) − 4 = 24

  • 44512Solution

    ((12 − 5) × 4) − 4 = 24

  • 44513Solution

    13 + ((4 × 4) − 5) = 24

  • 44611Solution

    ((11 − 6) × 4) + 4 = 24

  • 44613Solution

    ((13 − 6) × 4) − 4 = 24

  • 4478Solution

    (7 × (8 − 4)) − 4 = 24

  • 441112Solution

    (11 + (4 ÷ 4)) + 12 = 24

  • 441213Solution

    13 + (12 − (4 ÷ 4)) = 24

  • 4559Solution

    ((5 × 4) − 5) + 9 = 24

  • 45610Solution

    (5 × (10 − 6)) + 4 = 24

  • 4579Solution

    (5 × (7 − 4)) + 9 = 24

  • 45710Solution

    (10 × (7 − 5)) + 4 = 24

  • 45711Solution

    (5 × (11 − 7)) + 4 = 24

  • 45912Solution

    (5 × (12 ÷ 4)) + 9 = 24

  • 45913Solution

    (5 × (13 − 9)) + 4 = 24

  • 4669Solution

    (9 × (6 − 4)) + 6 = 24

  • 46612Solution

    (6 × (6 − 4)) + 12 = 24

  • 46710Solution

    (7 × (6 − 4)) + 10 = 24

  • 46810Solution

    ((10 − 6) × 4) + 8 = 24

  • 47711Solution

    (7 × (11 − 7)) − 4 = 24

  • 47810Solution

    (7 × (8 ÷ 4)) + 10 = 24

  • 47811Solution

    ((11 − 7) × 4) + 8 = 24

  • 47812Solution

    (7 × (12 − 8)) − 4 = 24

  • 47913Solution

    (7 × (13 − 9)) − 4 = 24

  • 48913Solution

    ((13 − 9) × 4) + 8 = 24

  • 491011Solution

    (10 × (11 − 9)) + 4 = 24

  • 4101012Solution

    (10 × (12 − 10)) + 4 = 24

  • 4101112Solution

    (10 + (12 ÷ 4)) + 11 = 24

  • 4101113Solution

    (10 × (13 − 11)) + 4 = 24

  • 5555Solution

    (5 × 5) − (5 ÷ 5) = 24

  • 5567Solution

    (5 × 5) − (7 − 6) = 24

  • 55611Solution

    ((11 − 5) × 5) − 6 = 24

  • 5588Solution

    (5 × 5) − (8 ÷ 8) = 24

  • 5589Solution

    (5 × (8 − 5)) + 9 = 24

  • 5599Solution

    (5 × 5) − (9 ÷ 9) = 24

  • 55910Solution

    (5 × 5) − (10 − 9) = 24

  • 551010Solution

    (5 × 5) − (10 ÷ 10) = 24

  • 551011Solution

    (5 × 5) − (11 − 10) = 24

  • 551013Solution

    13 + (10 + (5 ÷ 5)) = 24

  • 551111Solution

    (5 × 5) − (11 ÷ 11) = 24

  • 551112Solution

    (11 + (5 ÷ 5)) + 12 = 24

  • 551213Solution

    13 + (12 − (5 ÷ 5)) = 24

  • 551313Solution

    (5 × 5) − (13 ÷ 13) = 24

  • 5668Solution

    (6 × (8 − 5)) + 6 = 24

  • 56612Solution

    ((12 − 6) × 5) − 6 = 24

  • 5679Solution

    (9 × (7 − 5)) + 6 = 24

  • 561012Solution

    (6 × (10 ÷ 5)) + 12 = 24

  • 57710Solution

    (7 × (7 − 5)) + 10 = 24

  • 57910Solution

    (5 × (10 − 7)) + 9 = 24

  • 571010Solution

    (7 × (10 ÷ 5)) + 10 = 24

  • 59912Solution

    (5 × (12 − 9)) + 9 = 24

  • 591013Solution

    (5 × (13 − 10)) + 9 = 24

  • 5101111Solution

    (11 + (10 ÷ 5)) + 11 = 24

  • 6669Solution

    (6 × (9 − 6)) + 6 = 24

  • 66611Solution

    (6 × (11 − 6)) − 6 = 24

  • 66710Solution

    (6 × (10 − 7)) + 6 = 24

  • 66712Solution

    (6 × (12 − 7)) − 6 = 24

  • 66811Solution

    (6 × (11 − 8)) + 6 = 24

  • 66813Solution

    (6 × (13 − 8)) − 6 = 24

  • 66910Solution

    (10 × (9 − 6)) − 6 = 24

  • 661013Solution

    (6 × (13 − 10)) + 6 = 24

  • 661112Solution

    (11 + (6 ÷ 6)) + 12 = 24

  • 661213Solution

    13 + (12 − (6 ÷ 6)) = 24

  • 6799Solution

    (9 × (9 − 7)) + 6 = 24

  • 671010Solution

    (10 × (10 − 7)) − 6 = 24

  • 6888Solution

    (8 × (8 − 6)) + 8 = 24

  • 68910Solution

    (9 × (10 − 8)) + 6 = 24

  • 681011Solution

    (10 × (11 − 8)) − 6 = 24

  • 69911Solution

    (9 × (11 − 9)) + 6 = 24

  • 691012Solution

    (9 × (12 − 10)) + 6 = 24

  • 691112Solution

    (6 × (11 − 9)) + 12 = 24

  • 691113Solution

    (9 × (13 − 11)) + 6 = 24

  • 6101013Solution

    (10 × (13 − 10)) − 6 = 24

  • 6101212Solution

    (6 × (12 − 10)) + 12 = 24

  • 6111112Solution

    (11 + (12 ÷ 6)) + 11 = 24

  • 6111213Solution

    (6 × (13 − 11)) + 12 = 24

  • 6121313Solution

    13 + (13 − (12 ÷ 6)) = 24

  • 77910Solution

    (7 × (9 − 7)) + 10 = 24

  • 771013Solution

    13 + (10 + (7 ÷ 7)) = 24

  • 771112Solution

    (11 + (7 ÷ 7)) + 12 = 24

  • 771213Solution

    13 + (12 − (7 ÷ 7)) = 24

  • 7889Solution

    (8 × (9 − 7)) + 8 = 24

  • 781010Solution

    (7 × (10 − 8)) + 10 = 24

  • 791011Solution

    (7 × (11 − 9)) + 10 = 24

  • 7101012Solution

    (7 × (12 − 10)) + 10 = 24

  • 7101113Solution

    (7 × (13 − 11)) + 10 = 24

  • 88810Solution

    (8 × (10 − 8)) + 8 = 24

  • 881012Solution

    (8 × (12 − 10)) + 8 = 24

  • 881013Solution

    13 + (10 + (8 ÷ 8)) = 24

  • 881112Solution

    (11 + (8 ÷ 8)) + 12 = 24

  • 881213Solution

    13 + (12 − (8 ÷ 8)) = 24

  • 991013Solution

    13 + (10 + (9 ÷ 9)) = 24

  • 991112Solution

    (11 + (9 ÷ 9)) + 12 = 24

  • 991213Solution

    13 + (12 − (9 ÷ 9)) = 24

  • 10101013Solution

    13 + (10 + (10 ÷ 10)) = 24

  • 10101112Solution

    (11 + (10 ÷ 10)) + 12 = 24

  • 10101213Solution

    13 + (12 − (10 ÷ 10)) = 24

  • 10121213Solution

    13 + (10 + (12 ÷ 12)) = 24

  • 10131313Solution

    13 + (10 + (13 ÷ 13)) = 24

  • 12131313Solution

    13 + (12 − (13 ÷ 13)) = 24

5Make a multiple of 24

Expand the target: 48 ÷ 2, 72 ÷ 3, 96 ÷ 4, or 144 ÷ 6. This asks you to recognize a larger number as a useful stopping point before dividing it down.

1277
Reveal solutionHide solution for level 5

(7 × 7 − 1) ÷ 2 = 24

Build 49, adjust it to 48, then halve it.

Browse all 14 problemsHide 14 problems · Level 5

Ordered by the four numbers. Select a hand to reveal one of its easiest solutions. Scroll the list to see every problem in this level.

  • 1277Solution

    ((7 × 7) − 1) ÷ 2 = 24

  • 151111Solution

    ((11 × 11) − 1) ÷ 5 = 24

  • 161113Solution

    ((13 × 11) + 1) ÷ 6 = 24

  • 171313Solution

    ((13 × 13) − 1) ÷ 7 = 24

  • 2588Solution

    ((8 × 5) + 8) ÷ 2 = 24

  • 35713Solution

    ((13 × 5) + 7) ÷ 3 = 24

  • 351212Solution

    ((12 × 5) + 12) ÷ 3 = 24

  • 36611Solution

    ((11 × 6) + 6) ÷ 3 = 24

  • 441010Solution

    ((10 × 10) − 4) ÷ 4 = 24

  • 48811Solution

    ((11 × 8) + 8) ÷ 4 = 24

  • 5101011Solution

    ((11 × 10) + 10) ÷ 5 = 24

  • 6889Solution

    ((8 + 8) × 9) ÷ 6 = 24

  • 69910Solution

    ((10 × 9) ÷ 6) + 9 = 24

  • 6111212Solution

    ((12 × 11) + 12) ÷ 6 = 24

6Less obvious complementary totals

There are many ways to subtract your way to 24. Some, such as 35 − 11 or 42 − 18, ask you to see a relationship between less familiar intermediate totals.

14711
Reveal solutionHide solution for level 6

(4 + 1) × 7 − 11 = 24

Make 35 first, then recognize that the remaining 11 brings it back to 24.

Browse all 55 problemsHide 55 problems · Level 6

Ordered by the four numbers. Select a hand to reveal one of its easiest solutions. Scroll the list to see every problem in this level.

  • 12911Solution

    (11 × (2 + 1)) − 9 = 24

  • 13811Solution

    (11 × 3) − (8 + 1) = 24

  • 13910Solution

    ((10 + 1) × 3) − 9 = 24

  • 131112Solution

    ((11 + 1) × 3) − 12 = 24

  • 131212Solution

    ((12 × 3) × 1) − 12 = 24

  • 131213Solution

    ((12 × 3) − 13) + 1 = 24

  • 14711Solution

    (7 × (4 + 1)) − 11 = 24

  • 14913Solution

    ((9 × 4) − 13) + 1 = 24

  • 15710Solution

    (7 × 5) − (10 + 1) = 24

  • 23813Solution

    ((13 + 3) × 2) − 8 = 24

  • 231111Solution

    ((11 × 3) − 11) + 2 = 24

  • 231313Solution

    (13 × 3) − (13 + 2) = 24

  • 24712Solution

    ((7 + 2) × 4) − 12 = 24

  • 241112Solution

    ((11 − 2) × 4) − 12 = 24

  • 2579Solution

    (7 × 5) − (9 + 2) = 24

  • 2789Solution

    ((9 + 7) × 2) − 8 = 24

  • 33311Solution

    (11 × 3) − (3 × 3) = 24

  • 35813Solution

    (8 × 5) − (13 + 3) = 24

  • 37913Solution

    (9 × 7) − (13 × 3) = 24

  • 38810Solution

    ((10 × 8) − 8) ÷ 3 = 24

  • 44713Solution

    (13 × 4) − (7 × 4) = 24

  • 4489Solution

    (9 × 4) − (8 + 4) = 24

  • 45813Solution

    ((8 × 4) − 13) + 5 = 24

  • 47911Solution

    (11 × (7 − 4)) − 9 = 24

  • 471113Solution

    (11 × 4) − (13 + 7) = 24

  • 48813Solution

    ((13 × 8) − 8) ÷ 4 = 24

  • 491111Solution

    (11 × 4) − (11 + 9) = 24

  • 4101011Solution

    (11 × 4) − (10 + 10) = 24

  • 4121212Solution

    (12 × (12 ÷ 4)) − 12 = 24

  • 55811Solution

    (8 × 5) − (11 + 5) = 24

  • 5669Solution

    (9 × 6) − (6 × 5) = 24

  • 571011Solution

    (7 × (10 − 5)) − 11 = 24

  • 571212Solution

    (12 × 7) − (12 × 5) = 24

  • 5888Solution

    (8 × 5) − (8 + 8) = 24

  • 58913Solution

    (9 × 5) − (13 + 8) = 24

  • 581212Solution

    (12 × (8 − 5)) − 12 = 24

  • 591011Solution

    (9 × 5) − (11 + 10) = 24

  • 5101013Solution

    ((13 × 10) − 10) ÷ 5 = 24

  • 66610Solution

    (10 × 6) − (6 × 6) = 24

  • 66711Solution

    (11 × 6) − (7 × 6) = 24

  • 66913Solution

    (13 × 6) − (9 × 6) = 24

  • 67711Solution

    (7 × 6) − (11 + 7) = 24

  • 671111Solution

    (7 × (11 − 6)) − 11 = 24

  • 6121213Solution

    ((13 × 12) − 12) ÷ 6 = 24

  • 78810Solution

    (10 × 8) − (8 × 7) = 24

  • 78811Solution

    (8 × (11 − 7)) − 8 = 24

  • 791212Solution

    (12 × 9) − (12 × 7) = 24

  • 88811Solution

    (11 × 8) − (8 × 8) = 24

  • 88912Solution

    (12 × 8) − (9 × 8) = 24

  • 88913Solution

    (8 × (13 − 9)) − 8 = 24

  • 891111Solution

    (11 × (11 − 8)) − 9 = 24

  • 8111212Solution

    (12 × (11 − 8)) − 12 = 24

  • 9111212Solution

    (12 × 11) − (12 × 9) = 24

  • 9121212Solution

    (12 × (12 − 9)) − 12 = 24

  • 10121212Solution

    (12 × 12) − (12 × 10) = 24

7Fractional constructions

The last level requires a non-whole-number intermediate. These solutions can feel backwards: an awkward fraction is exactly what makes the puzzle work.

3388
Reveal solutionHide solution for level 7

8 ÷ (3 − 8 ÷ 3) = 24

8 ÷ 3 is 2⅔. Subtract that from 3 to get ⅓. Dividing 8 by ⅓ gives 24.

Browse all 16 problemsHide 16 problems · Level 7

Ordered by the four numbers. Select a hand to reveal one of its easiest solutions. Scroll the list to see every problem in this level.

  • 1346Solution

    6 ÷ (1 − (3 ÷ 4)) = 24

  • 1456Solution

    4 ÷ (1 − (5 ÷ 6)) = 24

  • 1555Solution

    5 × (5 − (1 ÷ 5)) = 24

  • 1668Solution

    6 ÷ (1 − (6 ÷ 8)) = 24

  • 181212Solution

    12 ÷ ((12 ÷ 8) − 1) = 24

  • 221111Solution

    11 × (2 + (2 ÷ 11)) = 24

  • 221313Solution

    13 × (2 − (2 ÷ 13)) = 24

  • 23512Solution

    12 ÷ (3 − (5 ÷ 2)) = 24

  • 241010Solution

    10 × (2 + (4 ÷ 10)) = 24

  • 25510Solution

    5 × (5 − (2 ÷ 10)) = 24

  • 27710Solution

    7 × (2 + (10 ÷ 7)) = 24

  • 3377Solution

    7 × (3 + (3 ÷ 7)) = 24

  • 3388Solution

    8 ÷ (3 − (8 ÷ 3)) = 24

  • 4477Solution

    7 × (4 − (4 ÷ 7)) = 24

  • 55711Solution

    5 × (7 − (11 ÷ 5)) = 24

  • 57711Solution

    7 × (5 − (11 ÷ 7)) = 24

Where the puzzles fall

Once every hand takes its easiest route, the distribution is concentrated near the beginning of the hierarchy. Select a count to browse all the problems at that level.

1,362 solvable hands, each counted once at its lowest level.
Solution levelHandsShare
1. Direct arithmetic226 16.59%
2. Obvious factors and cancellation226 16.59%
3. Construct familiar factors582 42.73%
4. Nearby totals and small corrections243 17.84%
5. Make a multiple of 2414 1.03%
6. Less obvious complementary totals55 4.04%
7. Fractional constructions16 1.17%
Total1,362100%

Level 3 is the largest group: 582 hands, or about 43%. Together, the first three levels account for 1,034 hands—roughly 76% of everything solvable. If I were building a practice sequence, I would spend a lot of time on familiar factors and the different ways to construct them.

Only 14 hands land in Level 5. That does not mean dividing a larger multiple is rarely possible. It means most hands with that route also have an easier one. The 2, 5, 8, 9 example is exactly that phenomenon.

At the far end, just 16 hands require fractions: about 1.17% of the solvable set. These are not merely hands with a clever fractional solution. Every solution to each of these hands passes through a non-integer value. Another example is 1, 3, 4, 6: 6 ÷ (1 − 3 ÷ 4) = 24.

What the classification does—and does not—say

Some boundaries are judgment calls. For this first pass, I treated subtracting 1 through 6 from an overshoot as a small correction. A mixed multiplication or division route requiring a larger subtraction falls in Level 6; that puts 36 − 12 there as well as 35 − 11. Those may not feel equally difficult to a particular player. The counts are reproducible under these rules, but the hierarchy is still open to refinement.

The preparation matters too. A solution does not become Level 3 just because its final step is 4 × 6 if getting the 6 required a harder division. Likewise, multiplying a difficult three-card solution by an unused 1 does not turn it into an easy cancellation puzzle.

These percentages describe distinct combinations, not the odds of a random deal. A hand with four different numbers can be dealt in more ways than four identical numbers. Each counts once here. The 458 unsolvable combinations are kept separate and receive no difficulty level.

A way to practice seeing

The useful next step is to separate recognition from calculation. A student who can multiply 6 by 4 may still need practice seeing that two differences can produce those factors. Another may recognize factor pairs immediately but never think to build 25 and subtract 1.

That suggests a more focused way to practice: choose a family of ideas, try several hands, then look for the same pattern in a mixed set. We can eventually compare these proposed levels with actual solve times and revise the order. For now, the most useful question to ask after finding an answer is: was there a simpler way to see 24?

Method and data

Analysis dated October 20, 2026. Numbers range from 1 to 13, repetitions are allowed, and each of the four cards must be used exactly once. Only +, −, ×, and ÷ are allowed; fractional and negative intermediates are permitted. There is no concatenation, exponentiation, or factorial.

The search used exact rational arithmetic and found 16,583 solution expressions after deduplicating operand swaps for addition and multiplication and interchangeable copies of a number. Different parenthesizations were retained. A separate enumeration of all distinct number orders, operator choices, and five binary tree shapes matched the solution sets for every hand. The classification selects the lowest level across all of a hand's solutions; it does not change the game's difficulty settings.

All 1,820 hands (CSV)Classification rules (JSON)Level counts (JSON)