- 99 → 30: −69: 99 → 30
- 30 → 50: +20: 30 → 50
- 50 → 107: +57: 50 → 107
The board hands you 99 and asks for 107; 3 of the tiles in the hand do it, in the right order.
The hand: +61, −69, +62, +20, +57. +61 moves you up by 61; from 99 that is 160. −69 takes 69 off; from 99 that is 30. +62 moves you up by 62; from 99 that is 161. +20 moves you up by 20; from 99 that is 119. +57 moves you up by 57; from 99 that is 156.
First, −69: 99 → 30. That leaves 77 still to gain. Move 2: +20, 30 → 50. That leaves 57 still to gain. +57 next, taking 50 to 107. This is the move that crosses 100.
Working backwards helps: the chain can finish on +57 or +20 or −69; take +57, which turns into 107 from 50. So the real question is how to reach 50 in 2 moves from 99 with the remaining tiles.
Step by step, the choices narrow: At 99, 5 tiles are legal and 3 of them still lead to 107. At 30, 4 tiles are legal and 2 of them still lead to 107. At 50, 3 tiles are legal and 1 of them still leads to 107.
Why not +61? Put it in for +57 and the chain ends on 111, 4 above the target — a near miss, but a miss. Why not +62? Put it in for +57 and the chain ends on 112, 5 above the target — a near miss, but a miss.
In total there are 12 ways to land within 10 of 107 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.