# 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:

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

Hint - Answer - Solution

## 42. Multiplicative sequence

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

- a
_{2} = 2 - a
_{mn} = a_{m}a_{n} for m, n relatively prime (multiplicative property)

Show that a_{n} = n, for every positive integer, n.

Hint - Solution

## 43. Sum of two powers

Show that n^{4} + 4^{n} 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 x^{y} + y^{x} > 1.

Hint - Solution

## 45. Area of regular 2^{n}-gon

Show that the area of a regular polygon with 2^{n} 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 (10^{9}) 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 x^{y} = y^{x}, 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) = a_{0} + a_{1}x + ... + a_{n}x^{n}, a_{0} is odd, and a_{0} + a_{1} + ... + a_{n} 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

Back to top

Nick Hobson

nickh@qbyte.org
*Last updated: March 15, 2003*