- 14 → 70: ×5: 14 → 70
- 70 → 87: +17: 70 → 87
- 87 → 99: +12: 87 → 99
- 99 → 178: +79: 99 → 178
14 on the left, 178 on the right, 4 moves between them. Here is one way to think it through.
The hand: +89, +12, +17, ×5, +79. +89 moves you up by 89; from 14 that is 103. +12 moves you up by 12; from 14 that is 26. +17 moves you up by 17; from 14 that is 31. ×5 multiplies by 5 — odd, so it keeps an odd value odd; 178 is not a multiple of 5. +79 moves you up by 79; from 14 that is 93.
First, ×5: 14 → 70. That leaves 108 still to gain. Move 2: +17, 70 → 87. That leaves 91 still to gain. Then +12: 87 becomes 99. That leaves 79 still to gain. +79 next, taking 99 to 178. It lands exactly on 178.
Working backwards helps: the chain can finish on +79 or +17 or +12; take +79, which turns into 178 from 99. So the real question is how to reach 99 in 3 moves from 14 with the remaining tiles.
Step by step, the choices narrow: At 14, 5 tiles are legal and 1 of them still leads to 178. At 70, 4 tiles are legal and 3 of them still lead to 178. At 87, 3 tiles are legal and 2 of them still lead to 178. At 99, 2 tiles are legal and 1 of them still leads to 178.
Why not +89? Put it in for +79 and the chain ends on 188, 10 above the target — a near miss, but a miss.
In total there are 6 ways to land within 10 of 178 without hitting it, and 6 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.