# Nick's Mathematical Puzzles: 61 to 70

## 61. Two cubes?

Let n be an integer.  Can both n + 3 and n2 + 3 be perfect cubes?

`Hint  -  Answer  -  Solution`

## 62. Four squares on a quadrilateral

Squares are constructed externally on the sides of an arbitrary quadrilateral.

Show that the line segments joining the centers of opposite squares lie on perpendicular lines and are of equal length.

`Hint  -  Solution`

## 63. Cyclic hexagon

A hexagon with consecutive sides of lengths 2, 2, 7, 7, 11, and 11 is inscribed in a circle.  Find the radius of the circle.

`Hint 1  -  Hint 2  -  Answer  -  Solution`

## 64. Balls in an urn

An urn contains a number of colored balls, with equal numbers of each color.  Adding 20 balls of a new color to the urn would not change the probability of drawing (without replacement) two balls of the same color.

How many balls are in the urn?  (Before the extra balls are added.)

`Hint  -  Answer  -  Solution`

## 65. Consecutive integer products

Show that each of the following equations has no solution in integers x > 0, y > 0, n > 1.

1. x(x + 1) = yn
2. x(x + 1)(x + 2) = yn
`Hint  -  Solution`

## 66. Quadratic divisibility

Show that, if n is an integer, n2 + 11n + 2 is not divisible by 12769.

`Hint  -  Solution`

## 67. Random number generator

A random number generator generates integers in the range 1...n, where n is a parameter passed into the generator.  The output from the generator is repeatedly passed back in as the input.  If the initial input parameter is one googol (10100), find, to the nearest integer, the expected value of the number of iterations by which the generator first outputs the number 1.  That is, what is the expected value of x, after running the following pseudo-code?

```n = 10100 x = 0 do while (n > 1) n = random(n) // Generates random integer in the range 1...n x = x + 1 end-do```
`Hint 1  -  Hint 2  -  Answer  -  Solution`

## 68. Difference of powers

Find all ordered pairs (a,b) of positive integers such that |3a − 2b| = 1.

`Hint  -  Answer  -  Solution`

## 69. Combinatorial sum

Find a closed form expression for

`Hint  -  Answer  -  Solution`

## 70. One degree

Show that cos 1°, sin 1°, and tan 1° are irrational numbers.

`Hint  -  Solution`