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Chain for

hard5 moves

The board

TargetTarget: 206
Running value: 27
  1. 27

0 of 5 moves

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

Show the solution
  1. 27 → 135: ×5: 27 → 135
  2. 135 → 15: ÷9: 135 → 15
  3. 15 → 65: +50: 15 → 65
  4. 65 → 129: +64: 65 → 129
  5. 129 → 206: +77: 129 → 206

Today's chain starts at 27 and has to finish on 206 in exactly 5 moves, each tile used once.

The hand: +64, +54, +77, ×5, +50, ÷9. +64 moves you up by 64; from 27 that is 91. +54 moves you up by 54; from 27 that is 81. +77 moves you up by 77; from 27 that is 104. ×5 multiplies by 5 — odd, so it keeps an odd value odd; 206 is not a multiple of 5. +50 moves you up by 50; from 27 that is 77. ÷9 only works on a multiple of 9; 27 is divisible by 9, and 206 is not.

First, ×5: 27 → 135. That leaves 71 still to gain. Then ÷9: 135 becomes 15. That leaves 191 still to gain. +50 next, taking 15 to 65. That leaves 141 still to gain. +64 next, taking 65 to 129. This is the move that crosses 100. Move 5: +77, 129 → 206. It lands exactly on 206.

Working backwards helps: the chain can finish on +50 or +77 or +64; take +77, which turns into 206 from 129. So the real question is how to reach 129 in 4 moves from 27 with the remaining tiles.

Step by step, the choices narrow: At 27, 6 tiles are legal and 2 of them still lead to 206. At 135, 5 tiles are legal and 1 of them still leads to 206. At 15, 4 tiles are legal and 3 of them still lead to 206. At 65, 3 tiles are legal and 2 of them still lead to 206. At 129, 2 tiles are legal and 1 of them still leads to 206.

Why not +54? Put it in for ×5 and the chain ends on 200, 6 below the target — a near miss, but a miss.

In total there are 30 ways to land within 10 of 206 without hitting it, and 12 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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    Chain for Saturday 12 September · puzzles