# St. Ives Problem/Rhind Papyrus Variant

## 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 $\ds$ $=$ $\ds 19 \, 607$

$\blacksquare$