Start at the answer
Four cards with no obvious route are hard to search forwards. Start at 24 instead and ask what makes it.
24 is 3 times 8. It is also 4 times 6, and 2 times 12. It is 25 minus 1, and 48 over 2.
Each of those is two parts. Look for two cards that make one part and two cards that make the other. With 3, 5, 6 and 8, the 3 and the 8 are already a pair, so 5 and 6 have to make 1. They do: 6 minus 5.
Find a useful grouping
Take 2, 4, 5 and 6. Aim at 4 times 6. The 4 and the 6 are on the table, so 2 and 5 have to leave them alone. They cannot, because every card is spent.
Aim at 24 as a sum instead. 2 times 5 is 10, and 4 plus 6 is 10. That is 20, which is short.
Aim at 4 times 6 again, but make the 4 and the 6 out of other cards. 6 stays as it is. 2 plus 5 is 7, which is not 4. Try 6 times 4 where the 4 comes from 6 minus 2: that spends 6 twice, and there is only one 6.
Now try 24 as 5 times 5 minus 1. 5 is on the table. 4 plus 2 is 6, not 5. Keep going: 5 times 6 is 30, and 30 minus 6 is 24. 6 comes from 4 plus 2. So 5 × 6 − (4 + 2) = 24.
When nothing whole works
Some deals have no route that stays in whole numbers. 1, 5, 5 and 5 is one of them. A pair can make 4 or 6, but no whole-number grouping uses all four cards to reach 24.
A fraction opens it. 1 over 5 is a fifth. 5 minus a fifth is 4 and four fifths. Five times that is 24.
So the answer is 5 × (5 − 1 ÷ 5). Distribute the outer 5: 5 × 5 − 5 × (1 ÷ 5) = 25 − 1 = 24.
When no whole route works, divide one small card by a larger one and see what the leftover fifth, third or quarter buys you.
Try it
You hold 1, 1, 4 and 9. Which pair should you make first?
- 9 − 1 and 4 − 1, to get 8 and 3
- 9 + 1 and 4 + 1, to get 10 and 5
Show the answer
9 − 1 and 4 − 1, to get 8 and 3
Correct. 9 minus 1 is 8 and 4 minus 1 is 3, and 8 times 3 is 24. Every card is spent once, so the deal is done.
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Practice Make 24