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