By Harold Reiter, posted February 27, 2017 —
This
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:
As
I was going to St. Ives,
I
met a man with 7 wives.
Each
wife had 7 sacks.
Each
sack had 7 cats.
Each
cat had 7 kits.
Kits,
cats, sacks and wives,
How
many were going to St. Ives?
The
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.
Because
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
7
+ 72 +
73 + 74.
Rather
than simply crunching the numbers on a calculator, a teacher might write this:
Let
S denote the sum we seek. Then
compute 7S. Next consider 7S – S:
7S
– S = 72 +
73 +
74 +
75 – (7
+ 72 +
73 +
74)
=
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)
This
leads to the surprisingly lovely formula
Now
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.
Similarly,
the finite geometric series
S = a + ar
+ ar2 +
... + arn
leads
to
rS – S
= r(a + ar + ar2 +
... + arn) – (a + ar + ar2 +
... + arn)
=
arn+1 –
a
and
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).
As
another extension, consider the series for which the nth term is n/2n. Multiply the infinite sum
by 1/2 to get
and
subtract to get
which
we recognize is just 2. Hence, our original sum is S = 4.
Even
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.
Harold
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.