- 38 → 83: swap digits: 38 → 83
- 83 → 19: −64: 83 → 19
- 19 → 133: ×7: 19 → 133
Start at 38. The target is 133, which is 95 away, and the budget is 3 moves.
The hand: swap digits, −64, ×7, −70, −56. swap digits turns 38 into 83 straight away, and it changes digit count when a number ends in 0. −64 takes 64 off; from 38 that is -26, which is below zero, so not yet. ×7 multiplies by 7 — odd, so it keeps an odd value odd; 133 is a multiple of 7. −70 takes 70 off; from 38 that is -32, which is below zero, so not yet. −56 takes 56 off; from 38 that is -18, which is below zero, so not yet.
First, swap digits: 38 → 83. That leaves 50 still to gain. Then −64: 83 becomes 19. That leaves 114 still to gain. Then ×7: 19 becomes 133. It lands exactly on 133.
Working backwards helps: the final move must be ×7, because 19 is the only value in reach that turns into 133. So the real question is how to reach 19 in 2 moves from 38 with the remaining tiles.
Step by step, the choices narrow: At 38, 2 tiles are legal and 1 of them still leads to 133. At 83, 4 tiles are legal and 1 of them still leads to 133. At 19, 1 tile is legal and 1 of them still leads to 133.
Why not −70? Play ×7, −64, −70 and the chain ends on 132, 1 below the target — a near miss, but a miss. Why not −56? It can be played, but every chain that uses it finishes away from 133.
In total there are 4 ways to land within 10 of 133 without hitting it, and exactly one order of the right tiles that does.
Once the set of tiles is right, the order usually follows from what is legal at each step.