Henry Ernest Dudeney/Modern Puzzles/92 - A Cow's Progeny/Solution

by : $92$

 * A Cow's Progeny

Solution

 * $121 \, 392$.

Proof
After $1$ year there will be $1$ cow.

After $2$ years there will be $2$ cows: the original cow and her first calf.

After $3$ years there will be $3$ cows: the original cow, her first calf and her second calf.

After $4$ years there will be $5$ cows: the original cow, her first calf and her second calf, her third calf and the first calf of her first calf.

This is the Rabbit Problem, but using a cow instead of a pair of rabbits.

By inspection, the total number of cattle after $n$ years is the $n + 1$th Fibonacci number $F_{n + 1}$.

So the total number of cattle after that time is $F_{26}$.

This can be looked up on, for example, which gives $121 \, 393$.

Because we are asked to count only the cow's descendants, we remove the original cow from the count, leaving $121 \, 392$.