Henry Ernest Dudeney/Puzzles and Curious Problems/36 - The Flagons of Wine/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $36$

The Flagons of Wine
A quart of Burgundy costs $4 \shillings 9 \oldpence$, but $3 \oldpence$ is returnable on the empty flagon,
so that the Burgundy seems to be worth $4 \shillings 6 \oldpence$
For $12$ of the capsules with which each of the quart flagons is sealed, a free flagon of the same value is obtained.
What is the value of a single capsule?
Obviously a twelfth of $4 \shillings 6 \oldpence$ which is $4 \tfrac 1 2 \oldpence$
But the free flagon also has a capsule worth $4 \tfrac 1 2 \oldpence$, so that this full flagon appears to be worth $4 \shillings 10 \tfrac 1 2 \oldpence$,
which makes the capsule worth a twelfth of $4 \shillings 10 \tfrac 1 2 \oldpence$, or $4 \tfrac 7 8 \oldpence$,
and so on ad infinitum, with an ever-increasing value.
Where is the fallacy, and what is the real worth of a capsule?


Solution

It is assumed that all empties are returned.

Then $13$ flagons of wine plus $1$ flagon and $1$ cap will cost $12 \times 4 \shillings 6 \oldpence$, less $3 \oldpence$ which makes $53 \shillings 9 \oldpence$ plus $12$ caps.

Without the capsule system, these $13$ flagons of wine would cost $13 \times 4 \shillings 6 \oldpence$ which is $58 \shillings 6 \oldpence$.

Therefore the difference, $4 \shillings 9 \oldpence$ represents the $12$ caps, each of which is worth $4 \tfrac 3 4 \oldpence$


For $12$ more flagons of wine plus $1$ flagon and $1$ cap, I pay $11 \times 4 \shillings 6 \oldpence$ less $3 \oldpence$ which is $49 \shillings 3 \oldpence$

The cost of these $12$, without caps, would be $54 \shillings$, which shows again the difference of $4 \shillings 9 \oldpence$, or $4 \tfrac 3 4 \oldpence$ a cap.

The cost of the caps will be the same for every dozen flagons of wine that we buy.

The stated error consists partly in overlooking the $3 \oldpence$, which is the value of the empty flagon when returned.


Sources

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