St. Ives Problem/Rhind Papyrus Variant
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Classic Problem
Problem $79$ of the Rhind Papyrus, written by Ahmes some time around $1650$ BCE, concerns:
- $7$ houses, each with:
- $7$ cats, each with:
- $7$ mice, each with:
- $7$ spelt, each with:
- $7$ hekat
Solution
Thus we have:
Houses | \(\ds 7 \) | ||||||||
Cats | \(\ds 7 \times 7 \) | \(\ds = \) | \(\ds 49 \) | ||||||
Mice | \(\ds 7 \times 7 \times 7 \) | \(\ds = \) | \(\ds 343 \) | ||||||
Spelt | \(\ds 7 \times 7 \times 7 \times 7 \) | \(\ds = \) | \(\ds 2401 \) | ||||||
Hekat | \(\ds 7 \times 7 \times 7 \times 7 \times 7 \) | \(\ds = \) | \(\ds 16807 \) |
As with the St. Ives Problem, the total can be calculated using the Sum of Geometric Sequence:
\(\ds \) | \(\) | \(\ds 7^1 + 7^2 + 7^3 + 7^4 + 7^5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 \times \paren {7^0 + 7^1 + 7^2 + 7^3 + 7^4}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7 \times \dfrac {7^5 - 1} {7 - 1}\) | Sum of Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds 19 \, 607\) |
$\blacksquare$
Sources
- c. 1650 BCE: Ahmes: Rhind Papyrus: Problem $79$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $7$
- 1992: David Wells: Curious and Interesting Puzzles ... (next): The World's Oldest Puzzle: $1$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $7$
- Weisstein, Eric W. "St. Ives Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StIvesProblem.html