## Brain-Teaser 418: Bell’s weights

**From The Sunday Times, 11th May 1969** [link]

“Now”, says Bell at the pub, “look intelligent and imaginative even if you don’t look beautiful by any means”. We swallow the insult and look solemn. Bell likes his jokes and we like his puzzles.

“Imagine you have some scales, but only three weights. However, you have a heap of Grade A sand, and a couple of bags; so you make two extra weights, one as heavy as all the first three put together, and the other twice as heavy as all the first three. Whereupon all the remaining sand is removed to a great distance”.

“With these five weights you must be able to weigh 1 ounce, 2 ounces, 3, 4, and so on, as far as possible. No gaps in that lot. It’s how far you can to the first gap I’m after. Usual prize — one pint for the best score before closing time”.

What should Bell’s five weights be to give the highest possible score?

This puzzle was included in the book **Sunday Times Brain Teasers** (1974, edited by Ronald Postill).

[teaser418]

## Jim Randell 10:54 am

on25 September 2022 Permalink |This Python program considers increasing values for the total of the first 3 weights, from 3 to 40.

It runs in 351ms.

Run:[ @replit ]Solution:The 5 weights are: 4, 6, 9, 19, 38.And this set of weights can be used to weigh all values from 1 to 64.

Using a set of balancing scales each weight can go in the left pan, right pan, or neither.

For for

nweights there are 3^npossibilities.But these include, not using any weights (all in neither pan), and each combination has a mirror image.

So the total maximum possible number of different weights would be:

For 5 weights we have:

W(5)= 121, and using a set consisting of increasing powers of 3 we can achieve this maximum and weigh all values between 1 and 121:But for a set of the form:

There are a 70 different arrangements. So the maximum number of different values we could achieve would be no more than this. And we can use the program to perform an exhaustive check up to this total, but there are no better solutions.

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