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committed e7d0ed1

Convert more single/double quotes to Unicode

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# File t2/toggle.html.wml

` #include '../template.wml'`
`+`
` <latemp_subject "Mathematical Analysis of the Toggle Squares Game" />`
` <latemp_more_keywords "Shlomi Fish, Toggle Squares, Ken Housley, Linear Algebra, Mathematics" />`
` `
` stuff in the text with HTML formatting.</p>`
` <p>I hope everything is clear to you. If you want clarification on`
` something, send a message to `
`-<mailto_link_to_self email="<main_email />" /> and I'll place`
`+<mailto_link_to_self email="<main_email />" /> and I’ll place`
` the clarification on this page.</p>`
` `
` <p>&nbsp;&nbsp;&nbsp;&nbsp; Shlomi Fish</p>`
` `
` <h2 id="One_Dimensional_Board">Analysis of a 1-D Toggle-board</h2>`
` <pre>`
`-First, I'll prove some analogous statements about a one-dimensional`
`+First, I’ll prove some analogous statements about a one-dimensional`
` toggle-squares board, i.e: conjectures that apply to the set of rows of size`
` 1*n and in which the presses are:`
` 110000 ,`
`       `
` Mathematically, it can be treated as vectors that belong to the linear`
` space (or vector field) {0,1}^n where the operations in the set {0,1} are`
`-XOR for '+' and AND for '*'.`
`+XOR for ‘+’ and AND for ‘*’.`
` `
` </pre>`
` `
` 111000...00 =`
` -------------`
` 001000...00`
`-and from here it can be proved inductively that the vectors with all 0's`
`+and from here it can be proved inductively that the vectors with all 0s`
` and one 1, are solveable. Since this is a basis of {1,0}^N then every state`
` is solveable. (i.e contained in Sp{Presses} )`
` `
` `
` <p>`
` A 1-D toggle field of size N is solveable for every state if N = 3k`
`-or N = 3k+1 and isn't if N = 3k+2 for some non-negative integer`
`+or N = 3k+1 and isn’t if N = 3k+2 for some non-negative integer`
` k.`
` </p>`
` `
` 110000`
` `
` and the same for every other column. Now, if we look on any row of cells,`
`-we will see that the "sticks" that were generated from the various columns`
`+we will see that the “sticks” that were generated from the various columns`
` in that row are also the 1-D TogSqrs sequence. Because N=3k,3k+1 again,`
` every row is solveable for all states, and therefore, the entire board is`
` solveable for all states.`
` `
` This provides us with a method to clear rows by pressing cells in the row`
` beneath them, and we can use it to clear all rows except the bottom one.`
`-Now, let's take a look at a situation in which there is a partially filled`
`+Now, let’s take a look at a situation in which there is a partially filled`
` row, and above it, at least two empty ones. I will prove that clearing it`
` by pressing the cells in the row above it, will make the two rows above it`
` a duplicate of its original state. For example if the status was:`
` meaning is that any number of them other than zero XORed together will`
` not generate the clear state.`
` `
`-I'll demonstrate on a 5*5 board:`
`+I’ll demonstrate on a 5*5 board:`
` `
`     |  |`
`   - 00000`
`   - 00000`
`     00000`
` `
`-The '-'s and '|'s mark the relevant presses. Let's assume that a`
`+The ‘-’s and ‘|’s mark the relevant presses. Let’s assume that a`
` certain number of presses in the square can form the clear state. If`
` so, then Pr(4,4) cannot be one of them because it is the only press`
` that can affect square (5,5). Moreover, Pr(3,4) and Pr(4,3) cannot be`
` which was returned from my matrix-canonization program.`
` `
` However, I believe it is agreeable that it is the simplest algorithm`
`-(yet) regarding "CPU" requirement and growing complexity, and has the`
`+(yet) regarding “CPU” requirement and growing complexity, and has the`
` advantage that it can be utilized without the aid of a computer.`
` `
` </pre>`