Arithmetic, Meet Algebra
By Harold Reiter, posted February 27, 2017 —
blog was inspired by a message from Roger Howe. The earliest known published
version of the riddle below comes from a manuscript dated to about 1730 (but
differs in referring to nine rather than seven wives). The modern form was
first printed circa 1825:
I was going to St. Ives,
met a man with 7 wives.
wife had 7 sacks.
sack had 7 cats.
cat had 7 kits.
cats, sacks and wives,
many were going to St. Ives?
rhyme is generally thought to refer to St. Ives, Cornwall, when it was a busy
fishing port and had many cats to stop the rats and mice from destroying the
fishing gear. Some people argue that it refers to St. Ives, Huntingdonshire, an
ancient market town.
only the riddle’s poser is “on his way” to St. Ives, the trick answer is 1. But
if we leave out the speaker, the intention is to ask the sum
+ 72 +
73 + 74.
than simply crunching the numbers on a calculator, a teacher might write this:
S denote the sum we seek. Then
compute 7S. Next consider 7S – S:
– S = 72 +
75 – (7
+ 72 +
75 + 74 – 74 +73 – 73 +72 – 72 – 7
75 – 7 = 7(74 – 1)
7(72 – 1)(72 + 1)
7 · 48 · 50 = 6 · 2800
So S = 2800. Adding the speaker, we get
2801 “things” going to St. Ives. Where is the algebra? So far, there isn’t any.
But replace the integer 7 with the letter x,
and consider xS – S:
xS – S = x2
+ x3 + x4 + x5 – (x + x2
+ x3 + x4)
= x5 + x4 – x4 + x3 – x3 + x2 – x2 – x
= x5 – x
= x(x4 – 1)
= x(x2 – 1)(x2 + 1)
leads to the surprisingly lovely formula
we have a solution to any of the possible variations of the riddle; we can
replace x with any value, say 9, as
in the original riddle. A teacher who is aware that students are eventually going
to need to understand the second general solution can pave the way by
discussing the specific case first.
the finite geometric series
S = a + ar
+ ar2 +
... + arn
rS – S
= r(a + ar + ar2 +
... + arn) – (a + ar + ar2 +
... + arn)
S = (ar n+1 –
a)/(r – 1).
In the case where ΙrΙ < 1, we have rn+1 → 0 as n→∞,
so the infinite series converges, and we have S = a/(1 – r).
another extension, consider the series for which the nth term is n/2n. Multiply the infinite sum
by 1/2 to get
subtract to get
we recognize is just 2. Hence, our original sum is S = 4.
a third-grade teacher with a firm math background can present the arithmetic in
a children’s poem in such a way that students later recognize the beautiful
ideas associated with a geometric series.
Reiter has taught mathematics for more than fifty-two years. In recent years,
he has enjoyed teaching at summer camps, including Epsilon, MathPath, and
MathZoom. His favorite current activity is teaching fourth and fifth graders
two days each week.
I really like this riddle. After watching my students work through several simplistic arithmetic and geometric word problems I think this problem would have lead to a better mathematical discussion and a better understanding of why the equations work the way they do. Thank you for sharing I'll definitely be saving this one.
This is a very interesting riddle that is a fun way of bringing math into any classroom where you have time to read a riddle and then give the students an opportunity to work it out. Like the article says even a third grade teacher with a strong math background could present this.