Re: Maths Puzzle
From: Jim Ferry (corklebath_at_hotmail.com)
Date: 06/29/04
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Date: 29 Jun 2004 12:37:11 -0700
coveringmynuts@hotmail.com (purinkle) wrote in message news:<a0dc8714.0406290144.5ff76c5a@posting.google.com>...
> This might be quite easy for some but can you help me with this puzzle please:
>
> A pile of x sticks lie on the floor.
> The keeper of the tower challenges you
> to take the last stick. You may take
> up to y sticks at a time. The keeper
> will then take up to y sticks. It is your
> turn first.
>
> Is there an easy way I cna work this out?
Suppose the value of y is fixed. Let's say y=3.
Now let's work out what happens for different
values of x.
We'll need some sort of notation. Let "i13", for
example, mean "It's my move, and there are 13 sticks
left". Similarly, let "k4" mean "It's the keeper's
move, and there are 4 sticks left."
For each of the games *states* (such as i13 or k4)
we want to determine if I "win" or "lose" assuming
both I and the keeper play optimally. So how do we
do this? Hmmm.
Well, let's start with the obvious:
i1, i2, i3: win
k1, k2, k3: lose
That is, if there are 1, 2, or 3 sticks left, I win
if it's my turn because I can take all the sticks.
Likewise, I lose if it's the keeper's turn (because
we're assuming he's not an idiot).
Now to something slightly less obvious:
i4: lose
k4: win
If the state of the game is i4 (my turn, 4 sticks), then
my three possible moves (take 1, 2, or 3 sticks) lead to
the states k3, k2, and k1 respectively. All of those
states are already labelled "lose". None of my choices
can lead to a win, so i4 is a state I wish to avoid.
Likewise, k4 is a state I'd love to see . . .
Therefore
i5, i6, i7: win
>From the states i5, i6, and i7, I can produce the state
k4, which I know wins. Of course, the keeper also knows
what to do on his turn if there are 5, 6, or 7 sticks, so
k5, k6, k7: lose
>From this it follows that
i8: lose
k8: win
At this point, you can see that a pattern is emerging.
You can guess that i4, i8, i12, i16, i20, etc, are all
going to be "lose" and that i(anything else) is going to
be a win.
Great! This is how mathematics is often done! You play
around with a problem until you see a pattern, then you
try to describe the pattern by making a conjecture:
* Conjecture: For y=3, I can win the game for all values
of x except those divisible by 4.
Now try to turn the conjecture into a theorem by proving it.
Go on! Get proving! You might trying letting P(i) be the
statement "The conjecture is true for 1 <= x <= 4i".
-- Jim Ferry at U of Illinois, Urbana-Champaign, (no, I don't educate) 2 email me l o o k up 1 row
- Next message: Donald G. Shead: "Re: The kilogram's average weight is about 8.9 N, the world around"
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- In reply to: purinkle: "Maths Puzzle"
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