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Subject: "turning turtles"     Previous Topic | Next Topic
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Conferences The CTK Exchange Middle school Topic #37
Reading Topic #37
stan stout (Guest)
guest
Feb-23-01, 03:56 PM (EST)
 
"turning turtles"
 
   i have been intrigued by teh turning turtles puzzle located in the games and puzzles section of this site. i would appreciate someone sharing the secret strategy to this puzzle. i would love to use this puzzle with my students! i am having difficulty making sense of the hints given in the puzzle. thanks so much!


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alexb
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672 posts
Feb-23-01, 04:25 PM (EST)
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1. "RE: turning turtles"
In response to message #0
 
   Let's assume you have mastered the game of Nim, because Truning Turtles is really Nim in disguise. The way it's presented at the site it's a Nim with eleven rows (or heaps) that contain 1, 2, 3, ..., 10, 11 checkmarks (or any other kind of items - pebbles, for example.)

O in the square #N means that the row has exactly N items. X in the square #N means that the row is empty.

To remove N items from row #N, click on the square #N.

To remove M < N items from row #N, click on squares #N and #(N-M). Why? There are two possibilites.

1. Square #(N-M) contains O. This means that row #(N-M) is empty.

2. Square #(N-M) shows X. This means that row #(N-M) is full, i.e. contains (N-M) items.

Forget for a moment about other rows. Your goal is to remove M items from row #N that at this point in the game contains N items. In other words you plan to leave a row with (N-M) items instead of N.

In the first case, you have row #N with N items and row #(N-M) empty. This means that square #N contains a O, while square #(N-M) contains X. After you click on these two squares, square #N will contain X, and square #(N-M) will contain O. This corresponds to row #N being empty and row #(N-M) being full with (N-M) items. Which means that from a configuration of two rows
empty/N you moved to a configuration (N-M)/empty, which is in fact what you wanted (except that along the way you swapped two rows, but who cares?)

In the second case, when you only look at rows #(N-M) and #N, the first one contains (N-M) elements, the second contains N elements. You wish to remove N elements from row #N leaving it with (N-M) elements. This would give you two rows of (N-M) elements each. But XOR on two equal numbers is 0. So leaving (N-M) items in row #N and having (N-M) items in row #(N-M) is the same as removing both rows. This is exactly equivalent to clicking on both boxes: #(N-M) and #N.

In class, you do not have to use 11 boxes. Take a smaller number. 3, 4, 5 will do very well.

For 3 boxes,

OOO is the same as Nim with 3 rows with number of elements being 1, 2, 3. OXO is Nim with 3 rows of 1, 0, 3 items, respectively.


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stan stout (Guest)
guest
Mar-09-01, 04:58 PM (EST)
 
2. "RE: turning turtles"
In response to message #1
 
   thanks! i can now beat it every time! i taught base 2 to my seventh graders then emailed them this website and told them to try the game. can't wait to hear from them and their parents!


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