- 72 → 36: ÷2: 72 → 36
- 36 → 41: +5: 36 → 41
- 41 → 130: +89: 41 → 130
- 130 → 126: −4: 130 → 126
- 126 → 169: +43: 126 → 169
Start at 72. The target is 169, which is 97 away, and the budget is 5 moves.
The hand: ÷2, +89, +56, +43, +5, −4. ÷2 only works on a multiple of 2; 72 is divisible by 2, and 169 is not. +89 moves you up by 89; from 72 that is 161. +56 moves you up by 56; from 72 that is 128. +43 moves you up by 43; from 72 that is 115. +5 moves you up by 5; from 72 that is 77. −4 takes 4 off; from 72 that is 68.
First, ÷2: 72 → 36. 72 divides by 2 exactly, which is why ÷2 is legal here. Then +5: 36 becomes 41. That leaves 128 still to gain. Then +89: 41 becomes 130. That leaves 39 still to gain. Then −4: 130 becomes 126. That leaves 43 still to gain. Move 5: +43, 126 → 169. It lands exactly on 169.
Working backwards helps: the chain can finish on −4 or +5 or +43 or +89; take +43, which turns into 169 from 126. So the real question is how to reach 126 in 4 moves from 72 with the remaining tiles.
Step by step, the choices narrow: At 72, 6 tiles are legal and 1 of them still leads to 169. At 36, 5 tiles are legal and 4 of them still lead to 169. At 41, 4 tiles are legal and 3 of them still lead to 169. At 130, 3 tiles are legal and 2 of them still lead to 169. At 126, 2 tiles are legal and 1 of them still leads to 169.
Why not +56? Play +89, +43, ÷2, +56, +5 and the chain ends on 163, 6 below the target — a near miss, but a miss.
In total there are 16 ways to land within 10 of 169 without hitting it, and 24 orders of the right tiles that do hit it.
Once the set of tiles is right, the order usually follows from what is legal at each step.