Why Chain has no order of operations

Chain never builds an expression. Each tile is applied to the running value at the moment you play it, and the result becomes the new running value. That single design decision removes BODMAS from the game entirely — there are no brackets to resolve, no multiplication that quietly happens before an addition, no precedence at all. What replaces precedence is order, and order is the thing most people get wrong on their first hard puzzle. If you want the rules themselves, the how to play page has them; this page is about the one rule that keeps costing tries.

An expression and a chain are different objects

Write 14 × 5 + 17 on paper and school arithmetic tells you to do the multiplication first, whatever order you wrote the symbols in. That convention exists because the expression is a static object and everyone has to read it the same way.

Chain has no static object. ×5 then +17 means: take the current value, multiply by five, and then take that and add seventeen. The tiles are functions applied one after another, and applying functions in sequence is associative but famously not commutative. Reading a chain left to right is not a convention you could have chosen differently — it is the only reading there is.

The practical upshot is that the notation you would use to write your solution down is misleading. 14 ×5 +17 +12 +79 is not an expression that evaluates to 178 under BODMAS; it is a recipe, and it happens to reach 178 because you follow it in the written order.

The trap: ×2 then +5 is not +5 then ×2

This is where the tries go. Take Sunday 6 September: start 59, target 298, four moves, hand +43 ×2 +93 +34 −37. The set is +93 +34 −37 ×2, and there are six working orderings — every arrangement of the three additive tiles, with ×2 last every single time.

The right chain: 59 + 93 = 152, + 34 = 186, − 37 = 149, × 2 = 298.

Play ×2 first instead and you get 118, then 211, then 245, then 208. That is 90 short, and 90 is not a coincidence: it is exactly the additive total you failed to double. The additions come to 93 + 34 − 37 = 90, and doubling them would have contributed another 90.

That generalises, and it is worth carrying: moving a ×k tile from the end of the chain to the start costs you (k − 1) times the sum of everything the multiplier no longer covers. With ×2 and additions worth 90, the gap is 90. With ×5 it would be four times that. So when a chain lands short by a suspiciously round amount, check whether your multiplier is in the wrong place before you start swapping tiles — the strategy guide treats a near miss as a wrong-tile signal, and this is the main case where it is a wrong-position signal instead.

Sometimes the multiplier has to go first

The opposite shape is just as common. Wednesday 9 September: start 14, target 178, four moves, hand +89 +12 +17 ×5 +79. Again six orderings work, and again the additions can appear in any order — but here ×5 is first in all six.

It is forced twice over. ×5 cannot be last, because the value before it would have to be 178 ÷ 5 = 35.6, and Chain runs on integers. It cannot be in the middle either: the smallest addition available is +12, and 26 × 5 = 130 leaves you needing 48 from two of +89 +17 +79, which is not on offer. +17 first gives 155 and needs 23, which is not on offer either, and +79 or +89 first pass the target the moment you multiply. So the only place left is the front: 14 × 5 = 70, then 87, 99, 178.

Two puzzles, two multipliers, and the correct position is opposite in each. There is no rule of thumb like "multiply early" that survives both. What survives is the question you should actually ask: is the multiplier covering the additions, or is it not? On 6 September the target is roughly twice a reachable number, so the ×2 covers everything. On 9 September the target is reachable by addition from a multiplied start, so the ×5 covers nothing.

What does commute

Not everything is a trap. Addition and subtraction commute with each other, always, as long as the running value stays inside 0 to 9999 on the way. Tuesday 1 September is the pure case: start 81, target 123, hand +69 +15 +9 −36 +8. The set is +69 +9 −36 and all six orderings reach 123, because no intermediate leaves the range. Monday 7 September behaves the same way.

That is why the strategy guide says Chain is a puzzle about the set rather than the order — on easy days, with only additive tiles in play, it is. The order only becomes a puzzle in its own right when a multiplier, a division or a swap enters the set, because those three are exactly the operations that do not commute with addition.

Division is a gate as well as an operation

÷n is refused unless the division is exact at that moment, so its position is constrained by arithmetic rather than by preference — and that constraint often pins down the rest of the chain.

Saturday 5 September: start 52, target 106, five moves, hand ÷4 −60 ÷9 +77 +9 ×6. Only two orderings work in the entire puzzle:

• ×6 −60 +9 ÷9 +77 — 312, 252, 261, 29, 106.
• ×6 +9 −60 ÷9 +77 — 312, 321, 261, 29, 106.

Everything is pinned except the two additive tiles, which commute with each other as usual. The reason is that ÷9 will only play on a multiple of 9, and the only one the chain can build is 261. ÷4 is legal on the start (52 ÷ 4 = 13) and leads nowhere; it is there to look reasonable.

Thursday 10 September shows the friendlier version. Start 36, target 117, hand ÷3 ×9 −28 +28 +31. After −28 +31 the value is 39, and from there both ÷3 ×9 and ×9 ÷3 land on 117 — 13 then 117, or 351 then 117. Multiplication and division by different numbers commute when both divisions happen to be exact, which here they are, because 39 and 351 are both multiples of 3. That is luck, not a law. Change one tile and one of those orders becomes an illegal move.

Swap digits barely commutes with anything

Swap is the tile that punishes order most. Friday 4 September: start 21, target 216, hand +70 ×8 −78 +60 +59 swap. Four orderings work and every one of them ends swap ×8 — the additive tiles shuffle freely, the last two moves do not move at all. The chain reaches 72, swaps to 27, and 27 × 8 = 216. Swap one move earlier, on 150, and you get 51; there is no way back.

Wednesday 2 September is subtler and worth studying. Start 54, target 415, hand +80 swap −6 −12 −10. Four orderings work:

• +80 swap −6 −10 — 134, 431, 425, 415.
• +80 swap −10 −6 — 134, 431, 421, 415.
• +80 −10 swap −6 — 134, 124, 421, 415.
• −10 +80 swap −6 — 44, 124, 421, 415.

Look at what −10 is allowed to do. It works before the swap and after it, and −6 only works after. The reason is arithmetic, not luck: reversing a three-digit number leaves the middle digit where it is, so subtracting 10 from 134 and then swapping gives the same answer as swapping and then subtracting 10. Subtracting 6 touches the units digit, which the swap moves to the front, so 134 − 6 = 128 swaps to 821 and the chain is gone. A tile commutes with swap exactly when it does not disturb the digits the swap relocates.

One order out of six

The hardest version of this is a puzzle where you have found the right set and still cannot solve it. Tuesday 8 September: start 97, target 516, three moves, hand −96 ×9 ×4 +54 +56. The set is +56 ×4 −96, and exactly one of its six orderings reaches the target:

• +56 ×4 −96 — 153, 612, 516.
• +56 −96 ×4 — 153, 57, 228.
• ×4 +56 −96 — 388, 444, 348.
• ×4 −96 +56 — 388, 292, 348.
• −96 +56 ×4 — 1, 57, 228.
• −96 ×4 +56 — 1, 4, 60.

Six orderings, four distinct endings, because the two additive tiles commute with each other and nothing else does. If you had the right three tiles and gave up after two wrong orders, the puzzle looked impossible and was not.

Checking an order before it costs you

A try is only recorded when you have played the full number of moves and missed. Undo and reset are free, and a refused tile is not a move at all. So the whole cost of being careful sits before the last tile, and that is where the checking belongs.

Three things to run before you commit the final move. Check the last digit of the target against the last digit your chain will produce — one multiplication or subtraction in the head, and it rules out most wrong orders instantly. Check that every division in the chain has a value it can divide exactly, at the position you are planning to play it. And check the swap: run the digits, and confirm the value it lands on is one the remaining tiles can finish from.

If all three pass and the chain still misses, then it really is the set that is wrong, and the strategy guide is where to go next. The FAQ covers what happens to your timer and your try count while you are experimenting — the short version is that thinking is free and only completed chains are counted.