Problems of the Month

April 2006's Problem:

Find a string of 132 consecutive composite natural numbers.

Solution:

There are infinitely many possibilities, this is one:

133! + 2, 133! + 3, 133! + 4, 133! + 5, ..., 133! + 133

March 2006's Problem:

Triangular numbers are so named because they can be arranged in a triangular configuration in a given pattern (see below).  The first few triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45 ...

Consider the following pattern:

1 = 1

1 + 5 = 6

1 + 7 + 7 = 15

1 + 9 + 9 + 9 = 28

1 + 11 + 11 + 11 + 11 = 45

...

This is a pattern that gives you the first few odd (the first, third, etc.) triangular numbers,

1) Generalize this pattern for all odd triangular numbers.  That is, find an expression for the nth term a n where n is odd.

2) Find a similar pattern for the even triangular numbers. (Hint: consider using zero to begin) and generalize the pattern by writing the nth term for the even triangular numbers.

3) Now give the formula for the nth triangular number.

Solution:

The following pattern gives the rest of the triangular numbers...

0 + 3 = 3

0 + 5 + 5 = 10

0 + 7 + 7 + 7 = 21

0 + 9 + 9 + 9 + 9 = 36

...

The pattern for odd triangular numbers generalizes, to a n = 1 + (n + 2)[(n - 1)/2]  for odd n .

The pattern for the even triangular numbers generalizes to a n = (n/2)(n + 1)  for even n .

The formula for the n th triangular number is a n = [n(n + 1)]/2

February 2006's Problem:

Starting with 4 colors, say  red, white, blue, and green, how many ways can the corners (vertices) of a square be colored?  Note a vertex is dimensionless, so that it can't actually be "colored."  It is more proper to say we are assigning colors to the vertices. Assume that reflections and rotations are allowed.

December 2005's Problem:

De Mere, an acquaintance of Pascal's gave him the following problem for which Pascal Proved the solution.  Which is greater, the probability of getting at least one six with four rolls of a single die, or the probability of getting at least one double-6 in 24 rolls of a pair of dice?

Find the radius of the circumscribed circle (figure not drawn to scale):

Solution:

[Last Updated: Sept 22, 2009]