A Kind of Solitaire

[davar55]

I recently came across this and thought it might be interesting here:

Start with a deck of T cards where T=n*(n+1)/2 is triangular.

Form a collection of piles of arbitrary sizes using up all T cards.

At each step, take one card away from every pile and with these
removed cards form a new pile.

Then:

(a) Eventually, the piles will have sizes {1,2,3,...,n} after which
each step cycles on this pattern.

(b) This pattern occurs within n*(n-1) steps.

(This is called Bulgarian solitaire and was introduced by Martin Gardner.)

Proofs of (a) and (b) exist, but you're welcome to try.

[retina]

It is a dull type of solitaire game. Since we know the outcome in advance and there is no strategy at all.

What actually is your puzzle here?

[davieddy]

Originally Posted by retina

What actually is your puzzle here?

To prove a) and b)

[Wacky]

Originally Posted by retina

What actually is your puzzle here?

I have to agree with this sentiment. It seems to me that there have been posted here a number of "mathematical exercises" and/or "conjectures" that we are being ask to solve in the name of a "puzzle".

These contrast highly from those examples such as the "hole in the sphere" or the logic problems that are characteristic of the problems posed by the "Mathematical Recreations" Greats of the last centuries.

[davieddy]

Originally Posted by Wacky

I have to agree with this sentiment. It seems to me that there have been posted here a number of "mathematical exercises" and/or "conjectures" that we are being ask to solve in the name of a "puzzle".

These contrast highly from those examples such as the "hole in the sphere" or the logic problems that are characteristic of the problems posed by the "Mathematical Recreations" Greats of the last centuries.

Well I am working on it with enthusiasm ATM.
I can see why 1,2,3,,,,,n stays as it is.
If Martin Gardner posed it, it qualifies for this thread IMHO.
What do you think of Nim?

David

[Wacky]

Originally Posted by davieddy

If Martin Gardner posed it, it qualifies for this thread IMHO. What do you think of Nim?

I will note that Martin Gardner presented many mathematically related topics which were not "puzzles" in the sense of Henry Dudeney or Sam Loyd.

In particular, I recall that he popularized John Conway's cellular automaton "Game of Life", as well as many other interesting topics.

As for Nim, I worked out the winning strategy decades ago. As a result, I find it of little more interest than Tic Tac Toe.

[davieddy]

Originally Posted by Wacky

As for Nim, I worked out the winning strategy decades ago. As a result, I find it of little more interest than Tic Tac Toe.

Do you want a "Bronze medal"

[davieddy]

Well I deem my "Nim" thread a success!
Is a proof of this solitaire puzzle beyond me?
It makes a nice little simulation program.

David

BTW in the light of Mally's demise, I regret my reference to a Bronze medal.