Chain strategy — how we find the set

Chain is a puzzle about the set, not the order. Once you know which tiles belong, the order is usually a matter of keeping the running value inside 0 to 9999, and most orders work. So every trick below is a way to rule tiles in or out before you play anything. Here is what we do, roughly in the order we do it, when a puzzle will not give. If you have not read the rules yet, start there.

Count what the tiles can reach

The first thing we do is add everything up. On an easy day with three moves and five tiles, the biggest number three additions can make is the start plus the three largest + tiles; the smallest is the start with the most negative three-tile combination the hand allows (two − tiles and the smallest +, if the hand has two). If the target sits outside that band, a multiplier or a swap has to be in the set, and that removes two tiles from consideration at once.

The medium puzzle from 9 September makes the point: start 14, target 178, hand +89 +12 +17 ×5 +79. The four additions total 197, so 14 + 197 = 211 is reachable — but only using all four, and the shortest solution uses exactly four tiles, so the set would have to be all four additions, which lands on 211, not 178. That is enough to know ×5 is in.

Read the last digit

Addition and subtraction move the last digit by the last digit of the tile. Multiplying by 2, 4, 6 or 8 always ends on an even digit; multiplying by 5 ends on 0 or 5. So look at the target's last digit and ask which combinations of tiles can produce it.

Target 160 with hand +65 −28 +55 +54 −36 (the easy puzzle from 7 September): the last digits of the tiles are 5, 8, 5, 4, 6, and the start ends in 9. Three tiles have to turn a 9 into a 0. 9 + 4 + 5 − 8 = 10 → last digit 0: that is the set +54 +55 −28, and it is the answer. 9 + 5 + 5 − 8 = 11 → 1, out. 9 − 6 + 5 + 4 = 12 → 2, out. Last digits alone leave one or two candidate sets on most easy days.

Test the division before you plan around it

÷n only plays when the running value is divisible by n. That is a strong constraint, because it has to be true at the moment you play the tile, not just at the start. Two habits follow from it.

First, check the start. If the start is divisible by the tile and none of the additions keeps it that way, the division goes first. The hard puzzle from 11 September (start 21, hand −37 +70 ÷3 +82 +84 +76) is exactly this: 21 ÷ 3 = 7 is legal, and 21 + 84 = 105 would also be divisible by 3, but 105 ÷ 3 = 35 leaves you needing 165 from +70 +76 +82 −37 in three moves, which no three of them make. Dividing first (21 → 7) opens the additions up.

Second, use multiplication to fix the last digit for a later division. If you hold ×4 and ÷2, playing ×4 guarantees the value is even; playing ÷2 afterwards is always legal. Puzzles on hard days like to hide the solution behind that kind of pairing.

Work backwards from the target

When forward reasoning stalls, undo the target. For every tile, ask what value would have to be on the board just before it: for +n that is target − n, for −n it is target + n, for ×n it is target ÷ n (only if exact), for ÷n it is target × n, and for swap digits it is the target's digits reversed. That gives you at most six "one move before the end" positions. Now do it again from each of those, and stop when you hit the start or a value that another tile can produce from the start.

Backwards from 178 in the medium puzzle: +79 needs 99, +17 needs 161, +12 needs 166, +89 needs 89, ×5 needs 35.6 — not an integer, so ×5 is not the last move. From 99: +12 needs 87, +17 needs 82, ×5 needs 19.8, +89 needs 10. From 87: +17 needs 70, and 70 is 14 × 5. That is the whole chain, found in three steps, without ever guessing.

When swap digits helps, and when it is bait

Swap is the only tile that changes the magnitude of a number without a multiplier. Watch for these shapes:

• The target is far above what the additions reach: the swap has to move the value into the hundreds. On 13 September (start 41, target 386, hand +15 swap digits +92 +40 +14) all four additions cap out at 202, so the swap must turn something in the 1xx range into a 3xx number. Two positions work: 133 → 331 (after +92, then +40 and +15 in either order) and 173 → 371 (after +92 +40, then +15). Swap before +92 and 81 becomes 18, which no two tiles recover.
• The running value ends in zero: swapping kills the trailing zero (120 → 21), which is a very fast way down. If the target is small and you hold ×n and a swap, multiply into a round number first.
• The value is a palindrome or a single digit: the swap is refused, because it would change nothing. If the only swap position you can find is on 44 or on 7, the swap is a decoy.

The decoy case is common. Every hand is built so each spare tile is playable somewhere and is part of at least one near miss, so a swap that "almost" works is not evidence that it belongs to the set.

Trust the near miss

A chain that lands within ten of the target usually means one wrong tile, not a wrong order. The puzzles are generated so that the spare tiles produce exactly these near misses — 170 instead of 160, 188 instead of 178, 385 instead of 386. When that happens, do not shuffle the order; look at the difference. 170 − 160 = 10, and the hand has +65 and +55, which differ by ten. Swap the one for the other.

If the difference is not something two tiles account for, the mistake is further back: either the multiplier is in the wrong place, or you have the wrong two additions.

Orders that stay in range

Once the set is right, the only way an order fails is by leaving 0 to 9999, or by asking for an inexact division. On easy days that almost never happens. On hard days, with a ÷n in the set, put the division where the value is divisible and keep the subtractions after the value is large enough not to go negative. Reordering costs nothing — undo is free — so when a legal order is not obvious, try the division and the multiplier early and the subtractions late.

How long it should take

We time ourselves loosely: a minute or two on Monday and Tuesday, three to five on medium days, and we stop feeling bad about Friday and Saturday somewhere around ten. The timer counts across tries and keeps going if you close the tab, so if you want an honest number, decide before you start whether you are solving it in one sitting.

None of this needs to be written down for the daily puzzle, but for the hard ones we do keep a scrap of paper with the last-digit table. If you find a trick we have not listed, the feedback button on this page goes straight to us, and the FAQ covers the practical questions — when the puzzle changes, what the site stores, and how to report a puzzle that looks wrong.