One Hundred Fowls/Bakhshali Version
Problem
$20$ men, women and childen earn $20$ coins between them as follows:
- Each man earns $3$ coins.
- Each woman earns $1 \frac 1 2$ coins.
- Each child earns $\frac 1 2$ a coin.
How many men, women and children are there?
Solution
- $2$ men
- $5$ women
- $13$ children.
Proof
Let $m$, $w$ and $c$ denote the number of men, women and children respectively.
Then we have:
\(\text {(1)}: \quad\) | \(\ds m + w + c\) | \(=\) | \(\ds 20\) | |||||||||||
\(\ds 3 m + 1 \frac 1 2 w + \frac 1 2 c\) | \(=\) | \(\ds 20\) | ||||||||||||
\(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 6 m + 3 w + c\) | \(=\) | \(\ds 40\) | clearing the fractions | |||||||||
\(\ds \leadsto \ \ \) | \(\ds 5 m + 2 w\) | \(=\) | \(\ds 20\) | $(2) - (1)$ |
We are of course subject to the condition that each of $m$, $w$ and $c$ is not fewer than $0$.
It is seen that $2 w$ is even.
Hence, in order for $5 m + 2 w$ also to be even, it is necessary for $m$ to be even.
If $m > 4$, then $5 m > 20$.
In that case $w < 0$ and so it must be that $m \le 4$.
This allows for $m$ to be equal to $0$, $2$ or $4$.
This gives the solutions:
\(\text {(1)}: \quad\) | \(\ds m\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds w\) | \(=\) | \(\ds 10\) | ||||||||||||
\(\ds c\) | \(=\) | \(\ds 10\) |
\(\text {(2)}: \quad\) | \(\ds m\) | \(=\) | \(\ds 2\) | |||||||||||
\(\ds w\) | \(=\) | \(\ds 5\) | ||||||||||||
\(\ds c\) | \(=\) | \(\ds 13\) |
\(\text {(3)}: \quad\) | \(\ds m\) | \(=\) | \(\ds 4\) | |||||||||||
\(\ds w\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds c\) | \(=\) | \(\ds 16\) |
However, let it be understood that there was at least one man, at least one woman and at least one child.
Then there is one solution remaining:
- $2$ men
- $5$ women
- $13$ children.
$\blacksquare$
Historical Note
If the Bakhshali Manuscript actually dates from as long ago as the $3$rd century C.E., this is probably the oldest instance of a One Hundred Fowls problem in the world.
Sources
- 1st Millennium: Anonymous: Bakhshali Manuscript
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Indian Puzzles: $47$