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

by : $14$

 * An Easy Settlement

Solution
There is not enough information to determine the answer uniquely.

This is the solution provided by :


 * 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 :


 * The threepenny piece: $\tfrac 1 4 \shillings = 3 \oldpence$
 * 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.