Henry Ernest Dudeney/Modern Puzzles/14 - An Easy Settlement/Solution

Modern Puzzles by Henry Ernest Dudeney: $14$

An Easy Settlement

Three men, Andrews, Baker and Carey, sat down to play at some game.

When they put their money on the table it was found that they each possessed $2$ coins only, making altogether $\pounds 1 \ 4 \shillings 6 \oldpence$

At the end of play Andrews had lost $5$ shillings and Carey had lost sixpence, and they all squared up by simply exchanging the coins.

What were the exact coins that each held on rising from the table?

Solution

There is not enough information to determine the answer uniquely.

This is the solution provided by Dudeney:

At the start of play

Andrews held a half-sovereign and a shilling,

Baker held a crown and a florin,

and Carey held a double florin and a half-crown.

After settlement,

Andrews held double florin and florin,

Baker the half-sovereign and half-crown,

and Carey held crown and a shilling.

Thus Andrews lost $5 \shillings$, Carey lost $6 \oldpence$, and Baker won $5 \shillings 6 \oldpence$

The selection of the coins is obvious, but their allotment requires a little judgment and trial.

However, there is more than one solution.

Solution $1$, as given by Dudeney

Andrews held:

$1$ half sovereign

$1$ shilling

for a total of $11$ shillings.

Baker held:

$1$ crown

$1$ florin

for a total of $7$ shillings.

Carey held:

$1$ double florin

$1$ half crown

for a total of $6 \tfrac 1 2$ shillings.

Hence the total of $11 + 7 + 6 \tfrac 1 2 = 24 \tfrac 1 2 \shillings = \pounds 1 \ 4 \shillings 6 \oldpence$

After the game:

Andrews held:

$1$ crown

$1$ shilling

for a total of $6$ shillings.

Baker held:

$1$ half sovereign

$1$ half crown

for a total of $12 \tfrac 1 2$ shillings.

Carey held:

$1$ double florin

$1$ florin

for a total of $6$ shillings.

Solution $2$, a slight variant on Dudeney

Andrews held:

$1$ half sovereign

$1$ double florin

for a total of $14$ shillings.

Baker held:

$1$ crown

$1$ florin

for a total of $7$ shillings.

Carey held:

$1$ shilling

$1$ half crown

for a total of $3 \tfrac 1 2$ shillings.

Hence the total of $14 + 7 + 3 \tfrac 1 2 = 24 \tfrac 1 2 \shillings = \pounds 1 \ 4 \shillings 6 \oldpence$

After the game:

Andrews held:

$1$ crown

$1$ double florin

for a total of $9$ shillings.

Baker held:

$1$ half sovereign

$1$ half crown

for a total of $12 \tfrac 1 2$ shillings.

Carey held:

$1$ shilling

$1$ florin

for a total of $3$ shillings.

Solution $3$, another variant on Dudeney

Andrews held:

$1$ half sovereign

$1$ half crown

for a total of $12 \tfrac 1 2$ shillings.

Baker held:

$1$ crown

$1$ florin

for a total of $7$ shillings.

Carey held:

$2$ half crowns

for a total of $5$ shillings.

Hence the total of $12 \tfrac 1 2 + 7 + 5 = 24 \tfrac 1 2 \shillings = \pounds 1 \ 4 \shillings 6 \oldpence$

After the game:

Andrews held:

$1$ crown

$1$ half crown

for a total of $7 \tfrac 1 2$ shillings.

Baker held:

$1$ half sovereign

$1$ half crown

for a total of $12 \tfrac 1 2$ shillings.

Carey held:

$1$ half crown

$1$ florin

for a total of $4 \tfrac 1 2$ shillings.

Solution $4$

Andrews held:

$1$ half sovereign

$1$ double florin

for a total of $14$ shillings.

Baker held:

$1$ crown

$1$ sixpence

for a total of $5 \tfrac 1 2$ shillings.

Carey held:

$1$ double florin

$1$ shilling

for a total of $5$ shillings.

Hence the total of $14 + 5 \tfrac 1 2 + 5 = 24 \tfrac 1 2 \shillings = \pounds 1 \ 4 \shillings 6 \oldpence$

After the game:

Andrews held:

$1$ crown

$1$ double florin

for a total of $9$ shillings.

Baker held:

$1$ half sovereign

$1$ shilling

for a total of $11$ shillings.

Carey held:

$1$ double florin

$1$ sixpence

for a total of $4 \tfrac 1 2$ shillings.

Proof

Recall the coins at the time of Dudeney:

The threepenny bit: $\tfrac 1 4 \shillings = 3 \oldpence$

The sixpence: $\tfrac 1 2 \shillings = 6 \oldpence$

The shilling

The florin: $2 \shillings$

The half crown: $2 \tfrac 1 2 \shillings = 2 \shillings 6 \oldpence$

The double florin: $4 \shillings$

The crown: $5 \shillings$

The half sovereign: $10 \shillings$

The sovereign: $20 \shillings$ or $\pounds 1$

For Andrews to settle with Baker, they need to exchange coins whose difference is $5 \shillings$

Hence Andrews had a half sovereign and Baker had a crown.

For Carey to settle with Baker, they need to exchange coins whose difference is $6 \oldpence$

Either:

Carey had a half crown and Baker had a florin.

or:

Carey had a shilling and Baker had a sixpence

Solutions $1$ to $3$

We add up:

$1$ half sovereign or $10 \shillings$

$1$ crown or $5 \shillings$

$1$ half crown or $2 \tfrac 1 2 \shillings$

$1$ florin or $2 \shillings$

gives $19 \tfrac 1 2$ shillings.

This leaves $5$ shillings, which can be made with either:

$2$ half crowns

or:

a double florin and a shilling

to give the total of $24 \tfrac 1 2 \shillings$

Baker starts the game with:

$1$ crown

$1$ florin

for a total of $7$ shillings.

However, the terms of the question place insufficient restriction on whether:

Andrews holds a double florin and Carey holds a shilling

Andrews holds a shilling and Carey holds a double florin

both Andrews and Carey hold a half crown.

Each of these gives rise to a different solution.

Hence before the game:

Andrews held:

$1$ half sovereign

Baker held:

$1$ crown

$1$ florin

Carey held:

$1$ half crown

After rising from the table:

Andrews held:

$1$ crown

Baker held:

$1$ half sovereign

$1$ half crown

for a total of $12 \tfrac 1 2$ shillings.

Carey held:

$1$ florin

and the ownership by Andrews and Carey of the other coins is undetermined.

Solution $4$

We add up:

$1$ half sovereign or $10 \shillings$

$1$ crown or $5 \shillings$

$1$ shilling

$1$ sixpence or $\tfrac 1 2$ a shilling

gives $16 \tfrac 1 2$ shillings.

This leaves $8$ shillings which can be made only with $2$ double florins, for a total of $24 \tfrac 1 2 \shillings$

Hence before the game:

Andrews held:

$1$ half sovereign

$1$ double florin

for a total of $14$ shillings.

Baker held:

$1$ crown

$1$ sixpence

for a total of $5 \tfrac 1 2$ shillings.

Carey held:

$1$ double florin

$1$ shilling

for a total of $5$ shillings.

After rising from the table:

Andrews held:

$1$ crown

$1$ double florin

for a total of $9$ shillings.

Baker held:

$1$ half sovereign

$1$ shilling

for a total of $11$ shillings.

Carey held:

$1$ double florin

$1$ sixpence

for a total of $4 \tfrac 1 2$ shillings.

$\blacksquare$

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $14$. -- An Easy Settlement