- 36 → 8: −28: 36 → 8
- 8 → 39: +31: 8 → 39
- 39 → 13: ÷3: 39 → 13
- 13 → 117: ×9: 13 → 117
Today's chain starts at 36 and has to finish on 117 in exactly 4 moves, each tile used once.
The hand: ÷3, ×9, −28, +28, +31. ÷3 only works on a multiple of 3; 36 is divisible by 3, and 117 is. ×9 multiplies by 9 — odd, so it keeps an odd value odd; 117 is a multiple of 9. −28 takes 28 off; from 36 that is 8. +28 moves you up by 28; from 36 that is 64. +31 moves you up by 31; from 36 that is 67.
First, −28: 36 → 8. That leaves 109 still to gain. Move 2: +31, 8 → 39. That leaves 78 still to gain. Then ÷3: 39 becomes 13. That leaves 104 still to gain. ×9 next, taking 13 to 117. It lands exactly on 117.
Working backwards helps: the chain can finish on ×9 or ÷3; take ×9, which turns into 117 from 13. So the real question is how to reach 13 in 3 moves from 36 with the remaining tiles.
Step by step, the choices narrow: At 36, 5 tiles are legal and 2 of them still lead to 117. At 8, 3 tiles are legal and 1 of them still leads to 117. At 39, 3 tiles are legal and 2 of them still lead to 117. At 13, 2 tiles are legal and 1 of them still leads to 117.
Why not +28? Put it in for +31 and the chain ends on 108, 9 below the target — a near miss, but a miss.
In total there are 17 ways to land within 10 of 117 without hitting it, and 4 orders of the right tiles that do hit it.
That is the whole solution; every other combination of tiles ends somewhere else.