# Nick's Mathematical Puzzles: 41 to 50

## 41. Crazy dice

Roll a standard pair of six-sided dice, and note the sum.  There is one way of obtaining a 2, two ways of obtaining a 3, and so on, up to one way of obtaining a 12.  Find all other pairs of six-sided dice such that:

1. The set of dots on each die is not the standard {1,2,3,4,5,6}.
2. Each face has at least one dot.
3. The number of ways of obtaining each sum is the same as for the standard dice.
`Hint  -  Answer  -  Solution`

## 42. Multiplicative sequence

Let {an} be a strictly increasing sequence of positive integers such that:

• a2 = 2
• amn = aman for m, n relatively prime (multiplicative property)

Show that an = n, for every positive integer, n.

`Hint  -  Solution`

## 43. Sum of two powers

Show that n4 + 4n is composite for all integers n > 1.

`Hint  -  Solution`

## 44. Sum of two powers 2

If x and y are positive real numbers, show that xy + yx > 1.

`Hint  -  Solution`

## 45. Area of regular 2n-gon

Show that the area of a regular polygon with 2n sides and unit perimeter is , where there are n − 1 twos under both sets of nested radical signs.

`Hint  -  Solution`

## 46. Consecutive subsequence

Given any sequence of n integers, show that there exists a consecutive subsequence the sum of whose elements is a multiple of n.
For example, in sequence {1,5,1,2} a consecutive subsequence with this property is the last three elements; in {1,−3,−7} it is simply the second element.

`Hint  -  Solution`

## 47. 1000 divisors

Find the smallest natural number greater than 1 billion (109) that has exactly 1000 positive divisors.  (The term divisor includes 1 and the number itself.  So, for example, 9 has three positive divisors.)

`Hint  -  Answer  -  Solution`

## 48. Exponential equation

Suppose xy = yx, where x and y are positive real numbers, with x < y.  Show that x = 2, y = 4 is the only integer solution.  Are there further rational solutions?  (That is, with x and y rational.)  For what values of x do real solutions exist?

`Hint  -  Answer  -  Solution`

## 49. An odd polynomial

Let p(x) be a polynomial with integer coefficients.  Show that, if the constant term is odd, and the sum of all the coefficients is odd, then p has no integer roots.  (That is, if p(x) = a0 + a1x + ... + anxn, a0 is odd, and a0 + a1 + ... + an is odd, then there is no integer k such that p(k) = 0.)

`Hint  -  Solution`

## 50. Highest score

Suppose n fair 6-sided dice are rolled simultaneously.  What is the expected value of the score on the highest valued die?

`Hint  -  Answer  -  Solution`