- 81 → 90: +9: 81 → 90
- 90 → 159: +69: 90 → 159
- 159 → 123: −36: 159 → 123
From 81 to 123: 3 moves, no tile twice, and only one set of tiles gets there.
The hand: +69, +15, +9, −36, +8. +69 moves you up by 69; from 81 that is 150. +15 moves you up by 15; from 81 that is 96. +9 moves you up by 9; from 81 that is 90. −36 takes 36 off; from 81 that is 45. +8 moves you up by 8; from 81 that is 89.
Working backwards helps: the chain can finish on −36 or +9 or +69; take −36, which turns into 123 from 159. So the real question is how to reach 159 in 2 moves from 81 with the remaining tiles.
First, +9: 81 → 90. That leaves 33 still to gain. +69 next, taking 90 to 159. That is 36 above the target, to be brought down. −36 next, taking 159 to 123. It lands exactly on 123.
Step by step, the choices narrow: At 81, 5 tiles are legal and 3 of them still lead to 123. At 90, 4 tiles are legal and 2 of them still lead to 123. At 159, 3 tiles are legal and 1 of them still leads to 123.
Why not +15? Put it in for +9 and the chain ends on 129, 6 above the target — a near miss, but a miss. Why not +8? Put it in for +9 and the chain ends on 122, 1 below the target — a near miss, but a miss.
In total there are 18 ways to land within 10 of 123 without hitting it, and 6 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.
A last check that costs nothing: 123 ends in 3, is odd, and divides by 3; the start, 81, ends in 1. Those two facts rule out most tile sets before you touch the board, which is the whole trick of Chain.