Rubber Band Man
looney lou’S rubber bAnd FACtory has, arguably, the least
efficient and most labor intensive method in the world for producing
rubber bands. lou makes them all by hand, taking two ends of a strip
of rubber, holding them in place on either side of his rubber fuser, and
then activating the fuser with a foot pedal. the fuser does a wonderful
job, and you can’t even see the seam. but each rubber band takes lou
a couple of minutes to make.
The rubber strips arrive on a truck
every Tuesday and are dumped into a
chute that deposits them in a big pile
on Lou’s work table. You would expect
that Lou’s process would be to pull out
a single strip of rubber, check the ori-
entation (to prevent Mobius bands),
What Lou actually does is sit with
the pile, pull any two loose ends out of
the pile, fuse them, and then let the new,
longer, band drop back into the stack to
work. He does this over and over until
there are no more loose ends anywhere
in the pile, then separates however many
bands were thus generated. In theory, if
he had 50 strips delivered, he could end
up with anywhere between one and 50
rubber bands at the end of this peculiar
process. (Of course, if he creates fewer
bands, they are each longer on average,
but that’s not the puzzle.)
This is the puzzle: If the truck delivers N strips each week, what is the
average number of rubber bands that
Lou will create each week? In addition, how many strips would need to
be delivered for Lou to average 10
bands per week?
Solutions may be emailed to the
Previous Issue’s Puzzle
author at cont.puzzles@gmail.
com. In order to make the solver
list, your solutions must by
received by May 31, 2012.
National Mathematics Year
As you may know, Srinivasa Ramanujan
was a brilliant mathematician who lived
a very brief life. I learned only recently
that his birthday (Dec. 22) is celebrated
in India as National Mathematics Day
and that India has declared 2012 to be
National Mathematics Year.
I’m not sure how National Mathematics Day is celebrated—it seems
like the perfect occasion for a pop quiz,
I suppose. And really, shouldn’t every
day be mathematics day—not just in India but throughout the world?
The story that most of you may remember about Ramanujan is that he could find
something interesting about any number.
It’s said that when the English mathematician G.H. Hardy visited Ramanujan in
taxi No. 1729 and commented that it was
a “boring number,” Ramanujan corrected
him by pointing out that 1,729 is the smallest natural number expressible in two
different ways as the sum of two cubes,
being both 13 + 123 and 93 + 103.
Back in the day, most of us would have
memorized the table of cubes through
some suitably high number and might
have been able to pull out the close connection to 1,728. But you have to admit that
Ramanujan’s comment was impressive—
especially as he was ill at the time. Based
upon this story, there’s a whole collection
of numbers now called “taxicab numbers.”
Inspired by this example, the following puzzle is actually a series of
1. The number 153 is the smallest number equal to the sum of the cubes of its
digits: 153 = 13 + 53 + 33 = 1 + 125 + 27.
There are three other numbers that are
equal to the sum of the cubes of their digits. Find them.
2. The number 1,634 is the smallest number equal to the sum of the fourth powers
of its digits: 1,634 = 14 + 64 + 34 + 44 = 1 +
1,296 + 81 + 256. There are two other numbers with this property. Find them.
3. The number 194,979 is the largest
number equal to the sum of the fifth
powers of its digits. There are five other
numbers with this property. Find them.
4. The number 145 is equal to the sum of
the factorials of its digits: 1! + 4! + 5! = 1 +
24 + 120. There is only one other number
with this property. Find it.
5. As far as I can tell, there’s only one
number equal to the sum of the sixth
power of its digits, but I’ve only checked
in the millions. Find it.
Many people pointed out that 0 and 1
are trivial solutions for the sum of pow-
ers questions. I thought those were
excluded, since each problem refer-
enced digits, plural; and a single number
is not normally referred to as a “sum.”
But fair enough.
1. Numbers equal to the sum of the
cubes of their digits: 153, 370, 371, 407.
Note for this one, you only need to
check for numbers up to 4* 9^ 3 = 2916.