## Brainteaser 1575: Quantum scales

**From The Sunday Times, 15th November 1992** [link]

The merchants of Mathematica only ever have to weigh things that weigh a whole number of “quantums”, that country’s unit of weight. They have a pair of balancing scales, and four weights. By using these weights (on either side of the balance) they can determine the weight of items weighing 1, 2, 3, 4, 5, … quantums, and they have designed the four weights so that this sequence continues as high as it possibly can with just four weights.

With their balance and weights they can determine the weight of a Maxpack, but not of anything heavier.

What is the weight of a Maxpack?

This puzzle was selected for the book *Brainteasers* (2002, edited by Victor Bryant). The wording above is taken from the book. It is slightly changed from the original puzzle, which removes an ambiguity in the original puzzle.

[teaser1575]

## Jim Randell 8:55 am

on26 May 2019 Permalink |First let’s consider a problem with being able to balance exact weights of

1 … n.If we can weigh the numbers

1 … nusingkweights, then by adding an extra weight of(2n + 1)we can weigh:So, with a weight of

1, we can weigh1 … 1.With weights of

(1, 3)we can weigh1 … 4.With weights of

(1, 3, 9)we can weigh1 … 13.With weights of

(1, 3, 9, 27)we can weigh1 … 40.(Each weight is an increasing power of 3, 3⁰ = 1, 3¹ = 3, 3² = 9, 3³ = 27, …).

And this would be the best we could manage if we had to balance exact weights.

But in this puzzle we know each object we have to weigh is one of the values

1 … N.So there is no point in being able to balance an object with a weight of 1, as if we can make a combination with a weight of 2, and determine that our object is lighter than it, then, as weights are quantised, it can only have a weight of 1.

In fact, there is no point in having weights of any odd value, as we determine that a weight is between

(2k)and(2k + 2), then it can only have a weight of(2k + 1).So, by doubling the 4 weights to

(2, 6, 18, 54)we can balance exact weights of2, 4, 6, 8, …, 80.And we will be able to determine the weight of any object weighing

1 … 80units:Solution:The weight of a Maxpack is 80 units.LikeLike