Bachet’s Four Weights Problem

Here’s another puzzle from Henry Dudeney’s article “The World’s Best Puzzles,” The Strand Magazine, December 1908. According to Dudeney, this puzzle is originally from Problèmes plaisans et délectables qui se font par les nombres (Pleasant and delectable number problems), by French mathematician Claude Gaspar Bachet de Méziriac (1851-1636).¹

A balance scale and four weights. Source: Internet Archive — A balance scale and four weights.

You have four (integral) weights $w_{1}, w_{2}, w_{3}, w_{4}$ and a balance scale such that you can weigh any (integral) number of pounds from 1 lb to 40 lbs. The weights may go on either side of the scale (e.g., with the object or in the opposite pan). What are the $w_{i}$ ?

This seems like a variation on the Frobenius coin problem, which, in general, is NP-hard. Fortunately, this specific instance is not.

As before, here’s Chirico’s The Mathematicians for you to look at while you try to solve this. Solution below.

The Mathematicians, Chirico (1917) source: WikiArt — The Mathematicians, Chirico (1917)

The Solution

The four weights are (1, 3, 9, 27). Before we confirm this computationally, let’s compute the solution using induction.

Let’s rephrase the problem as

You have $n$ weights… such that you can weigh any number of pounds in the range 1:m….”

where $m$ depends on how many weights you have. We can also observe that the sum of all the weights must equal the weight of the heaviest object you can measure, and that object must weigh $m$ pounds.

Then, for the case $n = 1$ , you have a single weight $w_{1} = 1$ , and you can weigh any object that weighs one pound (the interval 1:1). Now let’s look at the case $n = 2$ .

1. The weights $(w_{1} = 1, w_{2} = 3)$ can weigh any object from 1 to 4 pounds.

I’ll use $x$ to denote the object to be weighed, and the notation $[{s e t_{1}} | {s e t_{2}}]$ to denote what’s on the left and the right side of the scale. I’ll always put $x$ in the right-hand pan.

$x = 1$ is weighed as $[{1} | {x}]$ . This can be written as $1 = x$ .

$x = 2$ is weighed as $[{3} | {x, 1}]$ . This can be written as $3 = x + 1$ , or $3 - 1 = x$ .

$x = 3$ is weighed as $[{3} | {x}]$ . This can be written as $3 = x$ .

$x = 4$ is weighed as $[{1, 3} | {x}]$ . This can be written as $1 + 3 = x$ .

You can see from the above that the general form of the solution is

$s_{1} w_{1} + s_{2} w_{2} = x$

where $(w_{1}, w_{2}) = (1, 3)$ and $s_{1} \in {- 1, 0, 1}$ . A positive coefficient means the weight is in the left pan, a negative one means it’s in the right pan, and zero means the weight isn’t used. This is essentially a “signed trinary” representation of $x$ . For two digits, signed trinary can represent $3^{2} = 9$ values.

Let’s just see what that looks like in R:

library(poorman)

signs = c(-1, 0, 1)

S = expand.grid (s1 = signs, s2 = signs)

knitr::kable(S)

s1	s2
-1	-1
0	-1
1	-1
-1	0
0	0
1	0
-1	1
0	1
1	1

Let’s take the linear combination of s1 and s2, using the weights (1, 3).

S = S |>
  mutate(x = 1 * s1 + 3 * s2)

knitr::kable(S)

s1	s2	x
-1	-1	-4
0	-1	-3
1	-1	-2
-1	0	-1
0	0	0
1	0	1
-1	1	2
0	1	3
1	1	4

You can confirm for yourself that using another pair of weights won’t necessarily give you 9 unique values.

We are only interested in the positive values for our problem. We can calculate that the number of positive values is

$(3^{2} - 1) / 2 = (9 - 1) / 2 = 4.$

In other words, the weights (1, 3) can weigh the values 1:4.

2. The case of three weights

How many values can we weigh with three weights, and what are they? It’s clear that $w_{1}$ and $w_{2}$ must be the same as above, since any value in the range 1:4 can be represented as $s_{1} * 1 + s_{2} * 3 + 0 * w_{3}$ . Now to find $w_{3}$ .

We know that $3^{3} = 27$ , which means that we can represent $(27 - 1) / 2 = 13$ positive values, with 13 being the maximum. So the three weights must sum to 13, which gives us $w_{3} = 13 - (1 + 3) = 9$ . The set of weights is (1, 3, 9).

3. Finally, the case of four weights

We can calculate that $3^{4} = 81$ , which corresponds to $(81 - 1) / 2 = 40$ positive values (surprise!). Therefore, we have that $w_{4} = 40 - (1 + 3 + 9) = 27$ . And so the set of weights we want is (1, 3, 9, 27), as stated above.

Now, let’s confirm it.

S = expand.grid (s1 = signs,
                 s2 = signs,
                 s3 = signs,
                 s4 = signs) |>
  mutate(x = 1 * s1 + 3 * s2 + 9 * s3 + 27 * s4) |>
  filter(x > 0)

# confirm we have the values 1:40
stopifnot(S$x == 1:40)

knitr::kable(S)

	s1	s2	s3	s4	x
42	1	0	0	0	1
43	-1	1	0	0	2
44	0	1	0	0	3
45	1	1	0	0	4
46	-1	-1	1	0	5
47	0	-1	1	0	6
48	1	-1	1	0	7
49	-1	0	1	0	8
50	0	0	1	0	9
51	1	0	1	0	10
52	-1	1	1	0	11
53	0	1	1	0	12
54	1	1	1	0	13
55	-1	-1	-1	1	14
56	0	-1	-1	1	15
57	1	-1	-1	1	16
58	-1	0	-1	1	17
59	0	0	-1	1	18
60	1	0	-1	1	19
61	-1	1	-1	1	20
62	0	1	-1	1	21
63	1	1	-1	1	22
64	-1	-1	0	1	23
65	0	-1	0	1	24
66	1	-1	0	1	25
67	-1	0	0	1	26
68	0	0	0	1	27
69	1	0	0	1	28
70	-1	1	0	1	29
71	0	1	0	1	30
72	1	1	0	1	31
73	-1	-1	1	1	32
74	0	-1	1	1	33
75	1	-1	1	1	34
76	-1	0	1	1	35
77	0	0	1	1	36
78	1	0	1	1	37
79	-1	1	1	1	38
80	0	1	1	1	39
81	1	1	1	1	40

And we are done. At this point, I’ll note that we’ve just re-invented a signed base-3 number system: digit $i$ (counting from zero) represents the value $3^{i}$ . I just happened to be writing it backwards.

Footnotes

Among Bachet’s accomplishments is a method of constructing magic squares, a way of solving indeterminate equations with continued fractions, the proof of Bézout’s Identity, and a Latin translation of Arithmetica by Diophantus (he of the Diophantine equations). Famously, Fermat’s last theorem was a scribbled margin note on his copy of this very translation.↩︎

The Solution

1. The weights (w1=1,w2=3) can weigh any object from 1 to 4 pounds.

2. The case of three weights

3. Finally, the case of four weights

Footnotes

1. The weights $(w_{1} = 1, w_{2} = 3)$ can weigh any object from 1 to 4 pounds.