Here's a challenge that you may wish to try:
can you express all the numbers from 1 to 100 using
an arithmetic combination of only four 4's?
The operations and symbols that are allowed are:
the four arithmetic operations (+,x,,/),
concatenation (44 is ok and uses up two 4's),
decimal points (using 4.4 is ok),
powers (using 4^{4} is ok),
square roots,
factorials (using 4! is ok),
and overbars for indicating repeating digits
(e.g., writing .4 with an overbar
would be a way of expressing 4/9).
Ordinary use of parentheses are allowed.
No digits other than 4 are allowed.
So for instance, here is a complicated way to write 70 using four fours:
4 + [44/Sqrt(.4overbar)].
This problem is sometimes called the four fours
problem. I'm not sure where it first originated but it was
popularized by Martin Gardner, among others.
Presentation Suggestions:
This puzzle makes an excellent extra credit problem.
Or, you might suggest it as a joint project for a
whole class to work on: have them post solutions on a
bulletin board as they find them.
For a computer science course it makes a
nice programming exercise in the language prolog.
The Math Behind the Fact:
Actually, all the numbers less than 113 can be constructed
in this fashion. While I won't spoil the fun and tell you
the answers, let me just say (from experience)
that the hardest numbers to express in four 4's are the
numbers 69 and 73.
These require especially clever combinations
of the operations above.
A difficult (and as far as I know unsolved)
mathematical challenge is to prove that the number 113
cannot be constructed using these operations.
It should be noted that there are many versions
of this problem that have floated around, differing only
in the sets of operations that are allowed. (For instance, 113 can be done if you allow arccos as a function.)
How to Cite this Page:
Su, Francis E., et al. "Four Fours Problem."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
