- 97 → 153: +56: 97 → 153
- 153 → 612: ×4: 153 → 612
- 612 → 516: −96: 612 → 516
From 97 to 516: 3 moves, no tile twice, and only one set of tiles gets there.
The hand: −96, ×9, ×4, +54, +56. −96 takes 96 off; from 97 that is 1. ×9 multiplies by 9 — odd, so it keeps an odd value odd; 516 is not a multiple of 9. ×4 multiplies by 4 — an even number, so it can never land on an odd target directly; 516 is a multiple of 4. +54 moves you up by 54; from 97 that is 151. +56 moves you up by 56; from 97 that is 153.
First, +56: 97 → 153. That leaves 363 still to gain. Then ×4: 153 becomes 612. That is 96 above the target, to be brought down. Move 3: −96, 612 → 516. It lands exactly on 516.
Working backwards helps: the final move must be −96, because 612 is the only value in reach that turns into 516. So the real question is how to reach 612 in 2 moves from 97 with the remaining tiles.
Step by step, the choices narrow: At 97, 5 tiles are legal and 1 of them still leads to 516. At 153, 4 tiles are legal and 1 of them still leads to 516. At 612, 3 tiles are legal and 1 of them still leads to 516.
Why not ×9? Play −96, +56, ×9 and the chain ends on 513, 3 below the target — a near miss, but a miss. Why not +54? Put it in for +56 and the chain ends on 508, 8 below the target — a near miss, but a miss.
In total there are 3 ways to land within 10 of 516 without hitting it, and exactly one order of the right tiles that does.
If you saw the key tile first, the rest of the chain is bookkeeping.