Mathematics problems...?

sageowl
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Posts: 15

Mathematics problems...? Dec 3, 2012 16:05:44 GMT

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Post by sageowl on Dec 3, 2012 16:05:44 GMT

So I really don't know if you like things like this, but I love to solve mathematics problems in my spare time, so I thought I'd share.
A fun problem I found most recently was this:

Players 1, 2, 3, …, n are seated around a table and each has a single penny. Player 1 passes a penny to Player 2, who then passes two pennies to Player 3. Player 3 then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternately passing one penny or two to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers n for which some player ends up with all n pennies.

(Taken from here)
The nice thing about problems like this is they require mathematical thinking, rather than mathematical knowledge.
My answer was if m is a non-negative integer,
n= {2^m+1,2^m+2} for all m. (n can = 1, as well, but that's the degenerate case)

So what do you guys think?

brackets Foal Posts: 36	Mathematics problems...? Dec 21, 2012 6:58:02 GMT Quote Select Post Deselect Post Link to Post Member Give Gift Back to Top Post by brackets on Dec 21, 2012 6:58:02 GMT Hmm... 2? From what I understand, the most number of penny one can get is 2... Unless if you mean that the table is round and the game goes on and on until it ends? Then it would be n+2, right?

sageowl
Foal

Posts: 15

Mathematics problems...? Dec 21, 2012 16:33:51 GMT

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Post by sageowl on Dec 21, 2012 16:33:51 GMT

Wow, I totally forgot to say that the table was round...
Basically, for some numbers, like 3 for example, the game will end, while other numbers, like 8, would result in an infinite loop. The question is asking that for what numbers will the game end.

Last Edit: Dec 21, 2012 16:55:49 GMT by sageowl

brackets
Foal

Posts: 36

Mathematics problems...? Dec 30, 2012 20:40:47 GMT

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Post by brackets on Dec 30, 2012 20:40:47 GMT

All the odd numbers?
So far I have tried quite a number of digits, and as long as they are above 1 it should be a game.
Been thinking about this a lot, but its quite hard...Thanks for the exercise~
And apologies for the late reply, have a Happy New Year!

sageowl
Foal

Posts: 15

Mathematics problems...? Dec 31, 2012 2:27:56 GMT

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Post by sageowl on Dec 31, 2012 2:27:56 GMT

Here's the first example which repeats: 7.
Since it alternates passing off 1 and 2, the odd-numbered steps pass a 1 to the next step, while the even-numbered steps pass a 2 to the next step.

We start with:

1) [1][1][1][1][1][1][1]

2) [0][2][1][1][1][1][1]
The first person is eliminated from the table.
3) [0][3][1][1][1][1]
Another elimination here
4) [2][2][1][1][1]

5) [2][0][3][1][1]

6) [2][2][2][1]

7) [2][2][0][3]
or
7b) [2][2][3]

Now this is a circle, so the last gives one to the first.

8) [3][2][2]

9) [1][4][2]

10) [1][3][3]

11) [3][3][1]

12) [2][4][1]

13) [2][2][3]

Notice that this is the same as step 7, and will pass a 1 like step 7, so this sequence will continue from 7-12 ad infinitum.

One is the degenerate example, she automatically starts with all of the pennies, and two to six will all end with one person having that many pennies. Most sequences (but not all) after this will end with a repeating pattern (such as this one).

Post by sageowl on Dec 3, 2012 16:05:44 GMT

Post by brackets on Dec 21, 2012 6:58:02 GMT

Post by sageowl on Dec 21, 2012 16:33:51 GMT

Post by brackets on Dec 30, 2012 20:40:47 GMT

Post by sageowl on Dec 31, 2012 2:27:56 GMT

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