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.

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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.

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

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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!

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

(The Fibonacci numbers are defined by the recurrence equation F_{1} = 1, F_{2} = 1, with F_{n} = F_{n−1} + F_{n−2}, for n > 2.)

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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'.

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Solve the equation = x.

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

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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.

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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.

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Nick Hobson

nickh@qbyte.org