Henry Ernest Dudeney/Modern Puzzles/92 - A Cow's Progeny/Solution
Modern Puzzles by Henry Ernest Dudeney: $92$
- A Cow's Progeny
- "Supposing," said my friend Farmer Hodge, "that cow of mine to have a she-calf at the age of two years,
- and supposing she goes on having the like every year,
- and supposing every one of her young to have a she-calf at the age of two years,
- and afterwards every year likewise, and so on.
- Now, how many do you suppose would spring from that cow and all her descendants in the space of twenty-five years?"
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 On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008): A000045, which gives $121 \, 393$.
Because we are asked to count only the cow's descendants, we remove the original cow from the count, leaving $121 \, 392$.
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $92$. -- A Cow's Progeny
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $175$. A Cow's Progeny