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       \(\displaystyle 7 \)          
Cats       \(\displaystyle 7 \times 7 \)   \(\displaystyle = \)   \(\displaystyle 49 \)             
Mice       \(\displaystyle 7 \times 7 \times 7 \)   \(\displaystyle = \)   \(\displaystyle 343 \)             
Spelt       \(\displaystyle 7 \times 7 \times 7 \times 7 \)   \(\displaystyle = \)   \(\displaystyle 2401 \)             
Hekat       \(\displaystyle 7 \times 7 \times 7 \times 7 \times 7 \)   \(\displaystyle = \)   \(\displaystyle 16807 \)             


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