The Virginia Tech Regional Math Exam is Sat Oct 22, from 9-11:30am in Shan 2440. Come if you’re interested.
Remember to write up a solution to one Putnam Problem over the course of the semester!
You can download this problem set and next week’s TNG slate at http://www.math.hmc.edu/~omar/putnam/.
Putnam Classic # 6
Tuesday, October 11, 2012
Cubes
F1: The faces of a cube are colored black or white. How many distinguishable ways are there to
color the cube? (Note: two cubes are indistinguishably colored if they are identical after change of
orientation.)
F2: A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so
that one face of the cube is contained in the base of the cone. What is the side-length of the cube?
(Putnam 1998)
F3: Can any of these numbers:
704, 7704, 77704, 777704, 7777704...
ever be a perfect cube?
F4: (a) Suppose you are given a 3 × 3 × 3 cube of cheese, and a large flat knife. Try to cut the cube
into 1 × 1 × 1 cubes using as few cuts as possible. (You may move pieces around between cuts so as
to maximize the benefit of each cut.) What’s the fewest number of cuts you need?
(b) What’s the fewest cuts needed for a 4 × 4 × 4 cube (to be cut into 1 × 1 × 1 cubes)?
F5:
(a) Let D and E be n × n matrices with real entries. If E is not the zero matrix, and DE = 0, can
D be invertible?
(b) Let A and B be different n × n matrices with real entries. If A3 = B 3 and A2 B = B 2 A, can
A2 + B 2 be invertible?
(Putnam 1991)
And for a complete disdain for variety . . .
F6: (Another nine-digit problem...) The product of nine positive single digit integers is 9!
=362880. Their sum is 45. However they are not (1,2,3,4,5,6,7,8,9). Find them.
(Hess)
Hints:
3. Last week we saw that squares mod 4 could only be 0 or 1. Find an n such that cubes mod n have only a limited number of possibilities.
4. (a) Clearly 6 cuts are enough, if you don’t move pieces around. Can you do better?
5. (b) Avoid expressing the matrix by its entries; instead use matrix algebra. What happens if you sum those equations?