Wythoff symbol
Imagine you and your friend are playing a game with a bunch of stones. You take turns picking up stones, but you can only pick up either one stone or two stones each time. Now, sometimes you might end up with more stones than your friend, and sometimes your friend might end up with more stones than you. When you have more stones, we'll call that position "P", and when your friend has more stones, we'll call that position "N".
Now, let's say we want to write down this game in a special way so that we can easily remember which positions are "winning" positions (where the player who moved last wins) and which positions are "losing" positions (where the player who moved last loses). We can do this using something called the Wythoff symbol, which is just two numbers written as (a | b).
The first number, "a", represents the number of stones you take on your first move. So in our game, you can take either one stone or two stones, so a can only be 1 or 2. The second number, "b", represents the difference between the number of stones you and your friend have at the start of the game. So if you have 5 stones and your friend has 3, b would be 2.
Now, to figure out whether a particular position is winning or losing, we just need to look at its Wythoff symbol. If the position is at (a | b), then we can calculate something called the "complement" of that position, which is just (b-a | b). If the complement is a "losing" position (assuming both players play perfectly), then the original position is a "winning" position. And if the complement is a "winning" position, then the original position is a "losing" position.
So for example, let's say we're at position (2 | 3). We can calculate its complement as (3-2 | 3) = (1 | 3). Now we need to check whether (1 | 3) is a winning or losing position. To do this, we calculate its complement as (3-1 | 3) = (2 | 3). But (2 | 3) is a winning position (you can take one stone and force your friend into a losing position), so that means (1 | 3) is a losing position. Therefore, (2 | 3) is a winning position.
Overall, the Wythoff symbol is just a way of encoding a game with two players and a set of rules (in our case, taking either one or two stones at a time) so that we can easily figure out which player has the advantage in each position.
Now, let's say we want to write down this game in a special way so that we can easily remember which positions are "winning" positions (where the player who moved last wins) and which positions are "losing" positions (where the player who moved last loses). We can do this using something called the Wythoff symbol, which is just two numbers written as (a | b).
The first number, "a", represents the number of stones you take on your first move. So in our game, you can take either one stone or two stones, so a can only be 1 or 2. The second number, "b", represents the difference between the number of stones you and your friend have at the start of the game. So if you have 5 stones and your friend has 3, b would be 2.
Now, to figure out whether a particular position is winning or losing, we just need to look at its Wythoff symbol. If the position is at (a | b), then we can calculate something called the "complement" of that position, which is just (b-a | b). If the complement is a "losing" position (assuming both players play perfectly), then the original position is a "winning" position. And if the complement is a "winning" position, then the original position is a "losing" position.
So for example, let's say we're at position (2 | 3). We can calculate its complement as (3-2 | 3) = (1 | 3). Now we need to check whether (1 | 3) is a winning or losing position. To do this, we calculate its complement as (3-1 | 3) = (2 | 3). But (2 | 3) is a winning position (you can take one stone and force your friend into a losing position), so that means (1 | 3) is a losing position. Therefore, (2 | 3) is a winning position.
Overall, the Wythoff symbol is just a way of encoding a game with two players and a set of rules (in our case, taking either one or two stones at a time) so that we can easily figure out which player has the advantage in each position.
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