Mathematics Problem of the Week
Fall 2004
POW 6: Number Building
You’ve
seen the problem: can you use four copies of 4 combined with concatenation (use
the symbol #) and the operators +, -, *, /, and ^ to express all of the numbers
from 1 to 10.
Now,
try this one: Consider extending this problem to expressing any number with d
copies of the digit d : one 1, two 2's,
three 3's, etc.
For
example, we can express 22 with two 2’s by concatenation: 2#2. We can express 273 by (3^3)#3.
Define
U(d), the minimum inexpressible number
for digit d, to be the smallest positive integer that cannot be
expressed using exactly d copies of d combined with the above operations.
1. In the recent presidential
debate, President Bush announced that his administration had discovered that 11
could be expressed with 3 copies of any digit. Can you determine how?
2. Senator Kerry followed
that up by stating that since 11 can be written with three 3's, U(3) > 11. Is he
on the right track?
3. To settle the problem,
what are U(1), U(2), U(3), U(4), U(5), U(6), U(7)?
Solutions
should be submitted to Dr. Maegan Bos’ mailbox in the Math office or sent via
e-mail to mbos@stlawu.edu
Presentation counts! The prize-winning entry will
be selected from all correct submissions, based on the clarity, creativity and
elegance of the solution.
Look
for the SLU POW on the Web at http://it.stlawu.edu/~math/activities/