- 56 → 65: swap digits: 56 → 65
- 65 → 122: +57: 65 → 122
- 122 → 167: +45: 122 → 167
- 167 → 99: −68: 167 → 99
- 99 → 198: ×2: 99 → 198
The board hands you 56 and asks for 198; 5 of the tiles in the hand do it, in the right order.
The hand: swap digits, +57, −68, ×2, +39, +45. swap digits turns 56 into 65 straight away, and it changes digit count when a number ends in 0. +57 moves you up by 57; from 56 that is 113. −68 takes 68 off; from 56 that is -12, which is below zero, so not yet. ×2 multiplies by 2 — an even number, so it can never land on an odd target directly; 198 is a multiple of 2. +39 moves you up by 39; from 56 that is 95. +45 moves you up by 45; from 56 that is 101.
First, swap digits: 56 → 65. That leaves 133 still to gain. Move 2: +57, 65 → 122. This is the move that crosses 100. +45 next, taking 122 to 167. That leaves 31 still to gain. Move 4: −68, 167 → 99. That leaves 99 still to gain. ×2 next, taking 99 to 198. It lands exactly on 198.
Working backwards helps: the final move must be ×2, because 99 is the only value in reach that turns into 198. So the real question is how to reach 99 in 4 moves from 56 with the remaining tiles.
Step by step, the choices narrow: At 56, 5 tiles are legal and 2 of them still lead to 198. At 65, 4 tiles are legal and 2 of them still lead to 198. At 122, 4 tiles are legal and 2 of them still lead to 198. At 167, 3 tiles are legal and 1 of them still leads to 198. At 99, 2 tiles are legal and 1 of them still leads to 198.
Why not +39? Play swap digits, +39, ×2, +57, −68 and the chain ends on 197, 1 below the target — a near miss, but a miss.
In total there are 15 ways to land within 10 of 198 without hitting it, and 5 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.