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Ed from St Peter's College noticed a
pattern:
Most of the counters landed on the same card but there was one
counter which landed on a different card. We repeated this game
three times and in total four counters landed on differnt cards. In
one instance the counters all finished on the same card. This
happens because every card has a different number, so they all land
on the same numbered cards, so that again would change their path
to the same card. From this we have gathered that counters do land
on the same card most times. :D
Tom from Wilson School developed a
strategy:
You could put all the high numbers near the start and the low
numbers nearer the end so that some will land on the end one some
may not, it would be best to do a mixture of aces at the end.
However if you did only aces near the end you would get all your
counters to the end. The last card would have to be 1 more than you
have to move. If it is one until the end you would have to put a
number two there; if you were two from the end, a number 3 and so
on.
Joshua from Chesham Preparatory School
had a similar idea and found a precise solution through trial and
error:
Solution for counters on different cards: 1. Random start but this
was not useful. 2. We found out that aces, twos and threes (low
cards) weren't helpful as they put the counters onto the same or
close cards. 3. We tried 2 different methods to resolve the problem
of having 1 counter per card. 4. Our successful method was: - the
same high number one after each other - then low numbers (which
would be skipped) We carried on this technique until none of the
counters could move on. List of cards: 4 tens, 4 aces, 2 twos, 4
nines, 2 twos, 3 threes, 4 eights, 3 fours, 1 three, 4 sevens, 1
four, 2 fives, 4 sixes and 2 fives. This method worked
successfully!
It is worth noting that if two counters end
up on the same spot after a few steps, they are going to end up in
the same place when the game is over. This might make the trial and
error a bit faster.
The Maths Club at King Solomon Academy
noted:
Our results have proven that it is possible to land all four
counters on the same final card. This we believe is due to the
starting cards, which are all odd or are all even. However we found
out that if you have a mixture of both odd and even in the first
four cards our results will be varied. Firstly, if the sequence of
the first four cards are odd, odd, even, even, then the three of
the counters will land on the same card and one will land on a
separate card. Secondly, we realised that if the first four cards
were odd, odd, odd, even then all of them will land on the same
card apart from one.
Strangely, we found out that our predictions were proven wrong as
we thought that if the four starting cards were odd, even, even,
even, three counters would land on the same card. The experiment
completely proved this theory wrong as the results show that all
cards landed on the same final card.
The way the Maths Club came up with
theories, tested them and then tried to further and improve is a
useful technique. Well done to you all!
Can anyone develop a more rigourous
approach?