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 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.
== Sources ==
* : Solutions: $14$. -- An Easy Settlement
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions