- 31 → 98: +67: 31 → 98
- 98 → 89: swap digits: 98 → 89
- 89 → 145: +56: 89 → 145
- 145 → 180: +35: 145 → 180
The board hands you 31 and asks for 180; 4 of the tiles in the hand do it, in the right order.
The hand: +56, +35, swap digits, +67, +64. +56 moves you up by 56; from 31 that is 87. +35 moves you up by 35; from 31 that is 66. swap digits turns 31 into 13 straight away, and it changes digit count when a number ends in 0. +67 moves you up by 67; from 31 that is 98. +64 moves you up by 64; from 31 that is 95.
First, +67: 31 → 98. That leaves 82 still to gain. Then swap digits: 98 becomes 89. The digits of 98 read backwards give 89. Move 3: +56, 89 → 145. This is the move that crosses 100. +35 next, taking 145 to 180. It lands exactly on 180.
Working backwards helps: the chain can finish on +67 or +35 or +56; take +35, which turns into 180 from 145. So the real question is how to reach 145 in 3 moves from 31 with the remaining tiles.
Step by step, the choices narrow: At 31, 5 tiles are legal and 2 of them still lead to 180. At 98, 4 tiles are legal and 1 of them still leads to 180. At 89, 3 tiles are legal and 2 of them still lead to 180. At 145, 2 tiles are legal and 1 of them still leads to 180.
Why not +64? Put it in for +56 and the chain ends on 188, 8 above the target — a near miss, but a miss.
In total there are 18 ways to land within 10 of 180 without hitting it, and 4 orders of the right tiles that do hit it.
Tomorrow's chain uses a fresh hand; the habits — check the last digit, check what divides — carry over.