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Nick's Mathematical Puzzles: 81 to 90

81. Digit transfer (3 star)

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 (2 star)

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:

... and so on.

Hint  -  Answer  -  Solution

83. Divisibility (4 star)

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

Hint  -  Answer  -  Solution

84. Missing digits (3 star)

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 (2 star)

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 (2 star)

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

Right triangle ABC; folded so that C lies on AB.

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

Hint  -  Answer  -  Solution

87. 2004 (3 star)

Evaluate 22004 (modulo 2004).

Hint  -  Answer  -  Solution

88. Nested radicals (3 star)

Solve the equation  sqrt(4 + sqrt(4 - sqrt(4 + sqrt(4 - x)))) = x.

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

Hint 1  -  Hint 2  -  Answer  -  Solution

89. Square digits (3 star)

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 (2 star)

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

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Nick Hobson
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Last updated: August 31, 2004