- 59 → 95: swap digits: 59 → 95
- 95 → 133: +38: 95 → 133
- 133 → 35: −98: 133 → 35
- 35 → 117: +82: 35 → 117
From 59 to 117: 4 moves, no tile twice, and only one set of tiles gets there.
The hand: +84, −98, +38, +82, swap digits. +84 moves you up by 84; from 59 that is 143. −98 takes 98 off; from 59 that is -39, which is below zero, so not yet. +38 moves you up by 38; from 59 that is 97. +82 moves you up by 82; from 59 that is 141. swap digits turns 59 into 95 straight away, and it changes digit count when a number ends in 0.
Working backwards helps: the chain can finish on +82 or −98 or +38; take +82, which turns into 117 from 35. So the real question is how to reach 35 in 3 moves from 59 with the remaining tiles.
First, swap digits: 59 → 95. That leaves 22 still to gain. Then +38: 95 becomes 133. That is 16 above the target, to be brought down. Then −98: 133 becomes 35. That leaves 82 still to gain. +82 next, taking 35 to 117. It lands exactly on 117.
Step by step, the choices narrow: At 59, 4 tiles are legal and 1 of them still leads to 117. At 95, 3 tiles are legal and 2 of them still lead to 117. At 133, 3 tiles are legal and 2 of them still lead to 117. At 35, 2 tiles are legal and 1 of them still leads to 117.
Why not +84? Put it in for +82 and the chain ends on 119, 2 above the target — a near miss, but a miss.
In total there are 5 ways to land within 10 of 117 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.