- 19 → 91: swap digits: 19 → 91
- 91 → 59: −32: 91 → 59
- 59 → 151: +92: 59 → 151
A short chain today — 3 moves from 19 to 151 — but the hand holds more tiles than you need.
The hand: ×2, −32, −27, swap digits, +92. ×2 multiplies by 2 — an even number, so it can never land on an odd target directly; 151 is not a multiple of 2. −32 takes 32 off; from 19 that is -13, which is below zero, so not yet. −27 takes 27 off; from 19 that is -8, which is below zero, so not yet. swap digits turns 19 into 91 straight away, and it changes digit count when a number ends in 0. +92 moves you up by 92; from 19 that is 111.
Working backwards helps: the chain can finish on +92 or −32; take +92, which turns into 151 from 59. So the real question is how to reach 59 in 2 moves from 19 with the remaining tiles.
First, swap digits: 19 → 91. The digits of 19 read backwards give 91. Then −32: 91 becomes 59. That leaves 92 still to gain. +92 next, taking 59 to 151. It lands exactly on 151.
Step by step, the choices narrow: At 19, 3 tiles are legal and 1 of them still leads to 151. At 91, 4 tiles are legal and 2 of them still lead to 151. At 59, 3 tiles are legal and 1 of them still leads to 151.
Why not ×2? Play swap digits, ×2, −32 and the chain ends on 150, 1 below the target — a near miss, but a miss. Why not −27? Put it in for −32 and the chain ends on 156, 5 above the target — a near miss, but a miss.
In total there are 5 ways to land within 10 of 151 without hitting it, and 2 orders of the right tiles that do hit it.
That is the whole solution; every other combination of tiles ends somewhere else.