The Four Fours Puzzle by ChatGPT (v. 5.5 thinking)
As the summer term came to an end, Nigel, in his teaching days, found that the puzzle below got a maths class interested very quickly, especially when tackled collaboratively in pairs, groups or competing teams.
The problem is simple to state. Can you make every whole number from 1 to 100 using exactly four 4s and standard arithmetic operations?
For example, 2 can be made as:
4/4 + 4/4 = 2
Every expression must contain exactly four 4s. The operations may include addition, subtraction, multiplication, division, brackets, square roots, powers, factorials, decimal points, recurring decimals and joining two 4s together to make 44.
In the solutions below, 4! means 4 factorial, so 4! = 24. A dot above a 4 means that the 4 recurs: for example, .4̇ means 0.4444..., and .44̇ also means 0.4444..., but uses two written 4s.
| 1 | 44/44 |
| 2 | 4 × (4/(4 + 4)) |
| 3 | (4 + 4 + 4)/4 |
| 4 | 4 + 4 × (4 − 4) |
| 5 | (4 + 4 × 4)/4 |
| 6 | 4 + (4 + 4)/4 |
| 7 | 44/4 − 4 |
| 8 | 4 + 4 + 4 − 4 |
| 9 | 4 + 4 + 4/4 |
| 10 | (44 − 4)/4 |
| 11 | 44/√(4 × 4) |
| 12 | (4 + 44)/4 |
| 13 | √4 + 44/4 |
| 14 | 4 + 4 + 4 + √4 |
| 15 | 4 + 44/4 |
| 16 | 4 + 4 + 4 + 4 |
| 17 | 4 × 4 + 4/4 |
| 18 | 44/√4 − 4 |
| 19 | 4! − 4/4 − 4 |
| 20 | 4 × (4 + 4/4) |
| 21 | 4! + 4/4 − 4 |
| 22 | √4 × 44/4 |
| 23 | (4 × 4! − 4)/4 |
| 24 | 4 + 4 + 4 × 4 |
| 25 | (4 + 4 × 4!)/4 |
| 26 | 4 + 44/√4 |
| 27 | 4 + 4! − 4/4 |
| 28 | 44 − 4 × 4 |
| 29 | 4 + 4! + 4/4 |
| 30 | 4 × (4 + 4) − √4 |
| 31 | 4! + (4 + 4!)/4 |
| 32 | 4 × 4 + 4 × 4 |
| 33 | 4 + 4! + √4/.4 |
| 34 | √4 + 4 × (4 + 4) |
| 35 | 4! + 44/4 |
| 36 | 44 − 4 − 4 |
| 37 | 4! + (√4 + 4!)/√4 |
| 38 | 44 − √4 − 4 |
| 39 | 44 − √4/.4 |
| 40 | 44 − √(4 × 4) |
| 41 | (.4 + 4 × 4)/.4 |
| 42 | √4 + 44 − 4 |
| 43 | 44 − 4/4 |
| 44 | 4 + 44 − 4 |
| 45 | 44 + 4/4 |
| 46 | 4 − (√4 − 44) |
| 47 | √4 × 4! − 4/4 |
| 48 | 4 × (4 + 4 + 4) |
| 49 | 4/4 + √4 × 4! |
| 50 | 4 + √4 + 44 |
| 51 | ((4! − √4)/.4) − 4 |
| 52 | 4 + 4 + 44 |
| 53 | 44 + 4/.4̇ |
| 54 | 44 + 4/.4 |
| 55 | (44/√4)/.4 |
| 56 | 4 × (4 × 4 − √4) |
| 57 | √4 − (√4 − 4!)/.4 |
| 58 | (44 − 4!)/4 |
| 59 | 4!/.4 − 4/4 |
| 60 | 44 + 4 × 4 |
| 61 | 4/4 + 4!/.4 |
| 62 | 4 × 4 × 4 − √4 |
| 63 | (44 − 4)/4 |
| 64 | (4 + 4) × (4 + 4) |
| 65 | (4 + 44)/4 |
| 66 | √4 + 4 × 4 × 4 |
| 67 | √4 + (√4 + 4!)/.4 |
| 68 | 4 + 4 × 4 × 4 |
| 69 | 4 + (√4 + 4!)/.4 |
| 70 | √4 + 4! + 44 |
| 71 | (4! + 4.4)/.4 |
| 72 | 4 + 4! + 44 |
| 73 | (√(.4̇) + √4 × 4!)/√(.4̇) |
| 74 | 4 + (4 + 4!)/.4 |
| 75 | (4 + √4 + 4!)/.4 |
| 76 | 4 × (4! − 4) − 4 |
| 77 | (√(4/.4̇))4 − 4 |
| 78 | 4 × (4! − 4) − √4 |
| 79 | 4! − (√4 − 4!)/.4 |
| 80 | 4 × (4 + 4 × 4) |
| 81 | (4 − 4/4)4 |
| 82 | √4 − 4 × (4 − 4!) |
| 83 | 4! − (.4 − 4!)/.4 |
| 84 | √4 × 44 − 4 |
| 85 | (4! + 4/.4)/.4 |
| 86 | √4 × 44 − √4 |
| 87 | 4 × 4! − 4/.4̇ |
| 88 | 44 + 44 |
| 89 | 4! + (√4 + 4!)/.4 |
| 90 | √4 + √4 × 44 |
| 91 | 4 × 4! − √4/.4 |
| 92 | 4 + √4 × 44 |
| 93 | 4 × 4! − √(4/.4̇) |
| 94 | √4 + 4 × 4! − 4 |
| 95 | 4 × 4! − 4/4 |
| 96 | √4 × (4 + 44) |
| 97 | 4/4 + 4 × 4! |
| 98 | 4 − (√4 − 4 × 4!) |
| 99 | 4.4/(.4̇ − .4) |
| 100 | 4 × (4! + 4/4) |
Some numbers are easy to construct, others bafflingly hard - tell them not to try working through 1-100 in order, go for the low-hanging fruit first.
A brief comment on the rendering that some might experience. Depending on your screen you might find the upper dot in 0.4' (=0.4444...) visually similar to 0.4 (=0.4000..). To check whether you have this issue look at numbers 53 and 54. At first glance the equations might look the same- but of course they cannot be. Only 53 has the upper dot. Once you recognise it here, then you can check for it in the other examples that contain that dot.
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