UT Math Club Problem of the Month
February 2010
Problem I (A Cheesy Problem) A cube of cheese with side length 3 inches is divided
into 27 smaller cubes of side length 1 inch, in the obvious way (into a 3
dimensional, 3 × 3 × 3 grid of 27 cubes). One day a mouse comes along
and eats one small cube of cheese. On each following day the mouse eats
another small cube of cheese, each time choosing to eat a cube that shared
a face with the cube of cheese eaten the day before. In other words, the
mouse moves from one small cube of cheese to an adjacent cube, each day
eating exactly one small cube. Since there are 27 small cubes of cheese,
the mouse will eat the last cube on the 27th day. Is it possible for the
mouse to eat the center small cube on the last day?
Problem II (A Onederful Problem) Given a positive integer n, we say that an arithmetic expression A is Onederful of order n, if A involves only the operations of addition, multiplication and exponentiation, if the number 1
occurs in A precisely n times, and if no other numbers occur in A at all.
For example, the expression (1 + 1)1+1+1 + 1 is a Onederful expression of
order 6, and the expressions 1 + 2, 1/1, and 0 − 1 are not Onederful at
all. We say that a number x can be expressed Onederfully in n terms if x
is the value of a Onederful expression of order n. Now define a function
f : N → N by setting f (n) to be the largest number that can be expressed
Onederfully in n terms.
(a) Can you find a simple expression for f (n)? If not, then...
(n)
(b) Does there exist a polynomial p(x) ∈ R[x] such that limn→∞ fp(n)
=
0? If so, can you determine the smallest degree that a polynomial
can have while satisfying this property? If not, then...
(c) Does there exist a number a > 1 such that limn→∞
can you determine the smallest such number a?
f (n)
an
= 0? If so,
Disclaimer: This problem is relatively open ended. The winning submission will be the one that most clearly and accurately describes the behavior
of the function f .
Submission Deadline: Wednesday, February 17 by 11:59 PM