| home      | people     | research     | publications    | seminars    | events     | contact

### Schools Mathematics Grand Challenge

NOTE: THERE WAS ORIGINALLY A MISTAKE IN PROBLEM 22, WHERE THE TERM 1/(2x3x4) WAS MISSING. WE WILL ACCEPT THE ANSWER TO BOTH THE ORIGINAL AND CORRECTED PROBLEM.

PROBLEM 22:

Tom says he has figured out a quick way to find the number A given by:

A = 1/(1x2x3) + 1/(2x3x4) + 1/(3x4x5) + 1/(4x5x6) + 1/(5x6x7) + ... + 1/(99999999x100000000x100000001)

When Tom works out 1/(1-4A) he gets a whole number. What is this whole number?

NOTE: The "..." means that we have not written down all the parts of the sum from 1/(5x6x7) to 1/(99999999x100000000x100000001), but you need to include them when you figure out A.

PROBLEM 23:

To answer this question you will need to read the tip of the week to find out what a geometric sequence is.

Jo is practising pool on an American 9-ball pool table, where the balls are numbered from 1 to 9. Jo picks a geometric sequence and, if there are balls on the table whose numbers are in that sequence, then Jo pots all of them. Jo keeps picking geometric sequences (and potting the corresponding balls) until all nine balls have been potted.

If Jo is very good, and never misses a shot, what is the least number of geometric sequences that Jo needs to pick in order to clear the table?

HINT: You may need to use sequences with a ratio that is a whole number or a fraction.

PROBLEM 24:

A monkey sat in front of a special typewriter that only contained keys for the letters A, L, G, O, R, I, T, H, M, S. The monkey sat at the typewriter for a long time and pressed the keys. At the end of this time, the pattern typed on the page was quite remarkable. In fact, the monkey had typed out a large number of 10 letter words, all different mis-spellings of the word ALGORITHMS, one after another. Moreover, each mis-spelling had all of all the letters A, L, G, O, R, I, T, H, M, S and every letter was in the wrong position. If the monkey had typed every mis-spelling of this type exactly once, and had typed nothing other than these mis-spellings, how many times was the letter A typed in total?

EXTRA "PROBLEM" 25:

There are no marks for this problem, but please let us know the number of your favourite problem by entering it as the answer to problem 25.