How to solve Hushbell

Weight × distance

Every bar hangs from its centre. Its hooks sit 1, 2 or 3 steps out on either side, and each bell pulls with its weight × distance. A bar rests level when both sides pull the same total. So a light bell far out can balance a heavy bell close in: 1×3 = 3×1.

A bar only tilts once every hook on it is filled. Until then it hangs still, so an empty hook tells you nothing. When a full bar tilts, its heavy end is marked in red; when it balances, it turns indigo.

Lower bars

A hook can carry another bar instead of a bell. The lower bar pulls on its hook with the weight of everything hanging under it, wherever the bells sit on it. Every bar, lower or upper, must rest level.

Knotted, wrapped and tied bells

Knotted bells have a red knot on their cord. They are already in place and can't be moved, but they still pull, so count them in. They are clues, not obstacles.

Wrapped bells are wrapped in paper and show a symbol instead of a number. A small sample bar at the top corner already balances and holds one wrapped bell: read it like a last hook to find out what the wrapped bell weighs.

Tied pairs are two small bells tied together with a red cord. They hang on one hook and weigh their sum, so don't judge them by their size.

1. Last hook

When every other hook on a bar is known, only one weight can level it.

Add up what each side already pulls. The difference has to come from the last hook, so the bell there weighs difference divided by distance. If that is not a whole number, or there is no such bell in the tray, one of the bells you are sure of is not right. Hang the bells you are certain of first, then let the last hook follow.

2. Reach

A bell that heavy, that far out, could never be balanced by the other side.

Work out the most the other side could ever pull, using the heaviest bells left in the tray on its farthest hooks. A 6 three steps out pulls 18; if the other side can reach only 6, the 6 must hang closer in. This rules out hooks for the heaviest bells and often places them straight away.

3. Count the tray

Only so many bells of each weight are left, and just as many hooks can take them.

Every bell in the tray is used exactly once. When two bells weigh 2 and only two hooks anywhere can take a 2, both hooks get a 2 and no other hook can. Once a weight has found its hooks, cross it off everywhere else.

4. Odd and even

Bells on odd hooks (1 and 3 out) must add up to an even number.

A hook two steps out always pulls an even amount (weight × 2). For the two sides to match, the pulls from hooks 1 and 3 out must make up an even difference too, which happens only when the bells on those odd hooks add up to an even number, on both sides together. A knotted odd bell on an odd hook needs another odd bell beside it.

5. Threes

Leave out the bells 3 out; what is left on this bar must divide by 3.

Hooks three steps out always pull a multiple of 3. So if you ignore them, the difference between the two sides from the hooks 1 and 2 out must also be a multiple of 3 (0 counts). Try the candidate bells for those hooks and keep only the ones that pass.

6. The bar below

The upper bar decides how much everything hanging below must weigh.

Before you touch a lower bar, ask the bar above it. If the upper bar has a knotted 6 one step out on the left and the lower bar hangs two steps out on the right, then 6×1 = ?×2: the whole lower bar must weigh 3. Now you know which bells can go down there: only a set that adds up to 3 and levels the lower bar on its own.

7. Balance it in your head

Try the remaining bells on this bar in your head: only a few ways keep it level.

When nothing simpler works, pick the bar with the fewest empty hooks and list the ways the bells left could fill it. Start with the heaviest bell: where can it go? Most arrangements tilt, and often just one keeps the bar level. Checking in your head saves the trial and error.