- 54 → 44: −10: 54 → 44
- 44 → 124: +80: 44 → 124
- 124 → 421: swap digits: 124 → 421
- 421 → 415: −6: 421 → 415
54 on the left, 415 on the right, 4 moves between them. Here is one way to think it through.
The hand: +80, swap digits, −6, −12, −10. +80 moves you up by 80; from 54 that is 134. swap digits turns 54 into 45 straight away, and it changes digit count when a number ends in 0. −6 takes 6 off; from 54 that is 48. −12 takes 12 off; from 54 that is 42. −10 takes 10 off; from 54 that is 44.
Working backwards helps: the chain can finish on −10 or −6; take −6, which turns into 415 from 421. So the real question is how to reach 421 in 3 moves from 54 with the remaining tiles.
First, −10: 54 → 44. That leaves 371 still to gain. +80 next, taking 44 to 124. That leaves 291 still to gain. Move 3: swap digits, 124 → 421. The digits of 124 read backwards give 421. Move 4: −6, 421 → 415. It lands exactly on 415.
Step by step, the choices narrow: At 54, 5 tiles are legal and 2 of them still lead to 415. At 44, 3 tiles are legal and 1 of them still leads to 415. At 124, 3 tiles are legal and 1 of them still leads to 415. At 421, 2 tiles are legal and 1 of them still leads to 415.
Why not −12? Put it in for −6 and the chain ends on 409, 6 below the target — a near miss, but a miss.
In total there are 6 ways to land within 10 of 415 without hitting it, and 4 orders of the right tiles that do hit it.
If you saw the key tile first, the rest of the chain is bookkeeping.
A last check that costs nothing: 415 ends in 5, is odd, and divides by 5; the start, 54, ends in 4. Those two facts rule out most tile sets before you touch the board, which is the whole trick of Chain.