- 21 → 80: +59: 21 → 80
- 80 → 150: +70: 80 → 150
- 150 → 72: −78: 150 → 72
- 72 → 27: swap digits: 72 → 27
- 27 → 216: ×8: 27 → 216
From 21 to 216: 5 moves, no tile twice, and only one set of tiles gets there.
The hand: +70, ×8, −78, +60, +59, swap digits. +70 moves you up by 70; from 21 that is 91. ×8 multiplies by 8 — an even number, so it can never land on an odd target directly; 216 is a multiple of 8. −78 takes 78 off; from 21 that is -57, which is below zero, so not yet. +60 moves you up by 60; from 21 that is 81. +59 moves you up by 59; from 21 that is 80. swap digits turns 21 into 12 straight away, and it changes digit count when a number ends in 0.
First, +59: 21 → 80. That leaves 136 still to gain. +70 next, taking 80 to 150. This is the move that crosses 100. Then −78: 150 becomes 72. That leaves 144 still to gain. swap digits next, taking 72 to 27. That leaves 189 still to gain. Then ×8: 27 becomes 216. This is the move that crosses 100.
Working backwards helps: the final move must be ×8, because 27 is the only value in reach that turns into 216. So the real question is how to reach 27 in 4 moves from 21 with the remaining tiles.
Step by step, the choices narrow: At 21, 5 tiles are legal and 2 of them still lead to 216. At 80, 5 tiles are legal and 2 of them still lead to 216. At 150, 4 tiles are legal and 1 of them still leads to 216. At 72, 3 tiles are legal and 1 of them still leads to 216. At 27, 2 tiles are legal and 1 of them still leads to 216.
Why not +60? Put it in for +70 and the chain ends on 208, 8 below the target — a near miss, but a miss.
In total there are 8 ways to land within 10 of 216 without hitting it, and 4 orders of the right tiles that do hit it.
If you saw the key tile first, the rest of the chain is bookkeeping.