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Chain for

hard5 moves

The board

TargetTarget: 106
Running value: 52
  1. 52

0 of 5 moves

Reach the target in exactly 5 moves. Nothing here is recorded — this day is over.

Show the solution
  1. 52 → 312: ×6: 52 → 312
  2. 312 → 321: +9: 312 → 321
  3. 321 → 261: −60: 321 → 261
  4. 261 → 29: ÷9: 261 → 29
  5. 29 → 106: +77: 29 → 106

From 52 to 106: 5 moves, no tile twice, and only one set of tiles gets there.

The hand: ÷4, −60, ÷9, +77, +9, ×6. ÷4 only works on a multiple of 4; 52 is divisible by 4, and 106 is not. −60 takes 60 off; from 52 that is -8, which is below zero, so not yet. ÷9 only works on a multiple of 9; 52 is not divisible by 9, and 106 is not. +77 moves you up by 77; from 52 that is 129. +9 moves you up by 9; from 52 that is 61. ×6 multiplies by 6 — an even number, so it can never land on an odd target directly; 106 is not a multiple of 6.

Working backwards helps: the final move must be +77, because 29 is the only value in reach that turns into 106. So the real question is how to reach 29 in 4 moves from 52 with the remaining tiles.

First, ×6: 52 → 312. This is the move that crosses 100. Then +9: 312 becomes 321. That is 215 above the target, to be brought down. Then −60: 321 becomes 261. That is 155 above the target, to be brought down. ÷9 next, taking 261 to 29. That leaves 77 still to gain. Move 5: +77, 29 → 106. It lands exactly on 106.

Step by step, the choices narrow: At 52, 4 tiles are legal and 1 of them still leads to 106. At 312, 4 tiles are legal and 2 of them still lead to 106. At 321, 2 tiles are legal and 1 of them still leads to 106. At 261, 2 tiles are legal and 1 of them still leads to 106. At 29, 1 tile is legal and 1 of them still leads to 106.

Why not ÷4? Play ÷4, +77, ÷9, +9, ×6 and the chain ends on 114, 8 above the target — a near miss, but a miss.

In total there are 16 ways to land within 10 of 106 without hitting it, and 2 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.

How Chain works

You start on one number and have to end on the target, using each tile in the hand at most once, in whatever order you like. Tiles add, subtract, multiply, divide — division only when it comes out exact — or swap the digits of the running value. The number of moves is fixed: it is the length of the shortest solution, so playing fewer never ends on the target and there is no longer route to take. A chain that uses all its moves and misses counts as a try, and the board goes back to the start; undo and reset are free. Exactly one set of tiles works, and the rest are there to look plausible.

The full rules, with worked examples · how to find the set faster

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