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Chain for

hard5 moves

The board

TargetTarget: 200
Running value: 21
  1. 21

0 of 5 moves

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

Show the solution
  1. 21 → 7: ÷3: 21 → 7
  2. 7 → 91: +84: 7 → 91
  3. 91 → 167: +76: 91 → 167
  4. 167 → 130: −37: 167 → 130
  5. 130 → 200: +70: 130 → 200

The board hands you 21 and asks for 200; 5 of the tiles in the hand do it, in the right order.

The hand: −37, +70, ÷3, +82, +84, +76. −37 takes 37 off; from 21 that is -16, which is below zero, so not yet. +70 moves you up by 70; from 21 that is 91. ÷3 only works on a multiple of 3; 21 is divisible by 3, and 200 is not. +82 moves you up by 82; from 21 that is 103. +84 moves you up by 84; from 21 that is 105. +76 moves you up by 76; from 21 that is 97.

First, ÷3: 21 → 7. That leaves 193 still to gain. Then +84: 7 becomes 91. That leaves 109 still to gain. Move 3: +76, 91 → 167. This is the move that crosses 100. Then −37: 167 becomes 130. That leaves 70 still to gain. +70 next, taking 130 to 200. It lands exactly on 200.

Working backwards helps: the chain can finish on +76 or +84 or −37 or +70; take +70, which turns into 200 from 130. So the real question is how to reach 130 in 4 moves from 21 with the remaining tiles.

Step by step, the choices narrow: At 21, 5 tiles are legal and 1 of them still leads to 200. At 7, 4 tiles are legal and 3 of them still lead to 200. At 91, 4 tiles are legal and 3 of them still lead to 200. At 167, 3 tiles are legal and 2 of them still lead to 200. At 130, 2 tiles are legal and 1 of them still leads to 200.

Why not +82? Put it in for +84 and the chain ends on 198, 2 below the target — a near miss, but a miss.

In total there are 36 ways to land within 10 of 200 without hitting it, and 18 orders of the right tiles that do hit it.

Once the set of tiles is right, the order usually follows from what is legal at each step.

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 Friday 11 September · puzzles