# Nick's Mathematical Puzzles: 81 to 90

## 81. Digit transfer

Find the smallest positive integer such that when its last digit is moved to the start of the number (example: 1234 becomes 4123) the resulting number is larger than and is an integral multiple of the original number.  Numbers are written in standard decimal notation, with no leading zeroes.

`Hint  -  Answer  -  Solution`

## 82. Consecutive heads

A fair coin is tossed repeatedly until n consecutive heads occur.  What is the expected number of times the coin is tossed?

For example, two consecutive heads could be obtained as follows:

• HH (two tosses)
• THH (three tosses)
• HTHH or TTHH (four tosses)

... and so on.

`Hint  -  Answer  -  Solution`

## 83. Divisibility

Find all integers n such that 2n − 1 is divisible by n.

`Hint  -  Answer  -  Solution`

## 84. Missing digits

Given that 37! = 13763753091226345046315979581abcdefgh0000000, determine, with a minimum of arithmetical effort, the digits a, b, c, d, e, f, g, and h.  No calculators or computers allowed!

`Hint  -  Answer  -  Solution`

## 85. Fibonacci nines

Does there exist a Fibonacci number whose decimal representation ends in nine nines?

(The Fibonacci numbers are defined by the recurrence equation F1 = 1, F2 = 1, with Fn = Fn−1 + Fn−2, for n > 2.)

`Hint  -  Answer  -  Solution`

## 86. Folded card

A piece of card has the shape of a triangle, ABC, with BCA a right angle.  It is folded once so that:

• C coincides with C', which lies on AB; and
• the crease extends from Y on BC to X on AC.

If BC = 115 and AC = 236, find the minimum possible value of the area of YXC'.

`Hint  -  Answer  -  Solution`

## 87. 2004

Evaluate 22004 (modulo 2004).

`Hint  -  Answer  -  Solution`

## 88. Nested radicals

Solve the equation  = x.

(All square roots are to be taken as positive.)

`Hint 1  -  Hint 2  -  Answer  -  Solution`

## 89. Square digits

A perfect square has length n if its last n (decimal) digits are the same and non-zero.  What is the maximum possible length?  Find all squares that achieve this length.

`Hint  -  Answer  -  Solution`

## 90. Powers of 2: rearranged digits

Does there exist an integral power of 2 such that it is possible to rearrange the digits giving another power of 2?  Numbers are written in standard decimal notation, with no leading zeroes.

`Hint  -  Answer  -  Solution`