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# Total Count-Ability

How many different answers can you find and justify for the nursery rhyme Going to St. Ives?

As I was going to St. Ives

I met a man with seven wives.

Kits, cats, sacks and wives,

How many were going to St. Ives?

This version of the familiar nursery rhyme first appeared in 18th century England, but its roots go back much further than that. A variation appears in Leonardo Fibonacci’s Liber Abaci from the early 13th century which reads:

Seven old women are on the road to Rome. Each woman has seven mules, each mule carries seven sacks, each sack contains seven loaves, with each loaf there are seven knives, and each knife is in seven sheaths. How many objects are there: women, mules, sacks, loaves, knives, sheaths?

Even this version is not the first appearance of this problem. The earliest evidence dates back to ancient Egypt where the Rhind papyrus (originally written c. 1800 BCE) records the solution to an unwritten problem. The solution clearly suggests a form of the St. Ives rhyme:

Houses                               7

Cats                                 49

Mice                                343

Sheaves of wheat            2401

Hekats of grain              16807

Total                            19607

Dominic Olivastro, in his book Ancient Puzzles, conjectures that this list may have been the recording, by the scribe Ahmes, of the answer to a common puzzle of the time. This puzzle may have been as follows: A man owned seven houses. In each house, there were seven cats. Each cat killed seven mice. Each mouse ate seven sheaves of wheat. Each sheaf had seven hekats of grain. Grain, wheat, mice, cats, houses: What was the man’s entire estate?

Evidently in ancient times, the problem was presented as an exercise to be solved. This is in contrast to the St. Ives rhyme, which is presented as a trick question. The answer is said to be one, because if the narrator was on his way to St. Ives, the man and his wives would have been coming from St. Ives (although this is assumed and not explicitly stated), thus only one person is going to St. Ives. This modern interpretation greatly limits the scope of the problem. The goal of Total Count-Ability is to extend both the modern and ancient perspectives by having students justify multiple solutions to the rhyme.

Solutions

Click the arrow below to view the solution.

Following are six solutions and their justifications. Your class may think of other possible solutions, and as long as they can be logically justified, they are acceptable.

How many are going to St. Ives?

• One: The narrator was on his way to St. Ives and met the man (and his wives) coming from St. Ives.

• Two: The narrator met the man on the road to St. Ives and they were both going the same direction. The man told the narrator about his wives and their sacks, but none of them were actually present.

• 2802: The narrator came upon the man and his wives who were traveling to St. Ives slowly, encumbered by their sacks full of cats and kittens. [2 men, 7 wives, 49 sacks, 343 cats, 2401 kits]

• 2753: As above, the narrator came upon the man and his wives, but the sacks are not alive, so why would they be counted? [2 men, 7 wives, 343 cats, 2401 kits]

• Nine: As above, the narrator came upon the man and his wives, but sacks, cats and kits don’t matter; only people should be counted. [2 men, 7 wives]

• 2800: The last line of the poem says, “Kits, cats, sacks and wives, how many were going to Ives?” It is not asking about the narrator or the man, merely about the kits, cats, sacks and wives. [7 wives, 49 sacks, 343 cats, 2401 kits]

### 2 Responses to Total Count-Ability

1. Fernando says:

A sack may be inside another one. In this case, both sacks may share the same seven cats. This situation can be replicated independently or recursively (the latter yielding a “Russian-dolls” configuration).
That means that each wife can have 7, 14, 21, 28, 35, 42 or 49 cats, and the overall number of cats may be any multiple of seven between 49 and 343 (42 additional answers).

2. […] get teachers started, this week’s blog puzzle is “Total Count-Ability.” This is a great problem that can be solved in at least five valid ways depending on the […]