Last week, in Number Sense 045, Awkward Goat and Billy Goat Gruff found that their new Butts operation was associative, and, since Butts had an identity element, inverses and was closed, it formed a group.
This week they decided to explore operations a bit more, and made up another one.
“What we've done so far is quite interesting,” said Billy Goat Gruff, “but I'm not sure I could do anything with it.”
“Well, sometime we just do things for fun,” replied Awkward Goat. “Let's play with this a bit more.”
“How about another operation?” asked Billy Goat Gruff.
“All right,” said Awkward Goat, “I propose repeated Butting.”
“It will need a better name than that,” said Billy Goat Gruff. “Let's call it Pushing.”
“We can use Green this time for the identity element,” said Awkward Goat. “That way it will not turn out the same as Butting.”
“I still want to use all the colors in each row and column,” said Billy Goat Gruff, “so Yellow Pushes Yellow must be Red.”
“Must be?” asked Awkward Goat.
“If it was Green, then there would be two reds in the last column,” said Billy Goat Gruff.
“Very logical,” agreed Awkward Goat. “Then the upper right and lower left corners must be Green.”
“And the last corner must be Yellow,” finished Billy Goat Gruff.
“Now we should check whether it is associative,” said Awkward Goat.
“Here are the six combinations using all three colors,” said Awkward Goat.
“I would guess that three of the same color would be associative as well,” said Billy Goat Gruff.
“It would be easy enough to prove, instead of guess,” said Awkward Goat.
“Well, if you have three of the same color, then two of them Push, and you get another color.”
“Except for Green,” reminded Awkward Goat.
“Except for Green,” said Billy Goat Gruff, “which is the identity element, so it should give Green when pushing another Green.”
“OK,” said Awkward Goat, “very good. Carry on.”
“As I was saying,” continued Billy Goat Gruff, “you get another color, and then push the original color. So whether you group the first two, or the last two, it will come out the same.”
“Hmmm,” said Awkward Goat. “I think you assume that Push is commutative.”
“Isn't it?” asked Billy Goat Gruff.
“Well, maybe,” said Awkward Goat, “but we haven't proven it yet. We haven't even considered it yet.”
“See here,” said Billy Goat Gruff. “Yellow Pushes Green is Yellow! Green Pushes Yellow is also Yellow! And so on! And so forth!”
“Yes, yes,” said Awkward Goat, “now that you point it out, I see it. Push is Commutative.”
“Therefore!” said Billy Goat Gruff:
“Where did those colors come from?” asked Awkward Goat.
“They are unknowns,” said Billy Goat Gruff. “It doesn't matter which color we start with, the point is they are all the same. Two of them Push, to give another color. Since Push is Commutative, the order doesn't matter, when you do the second push, you get the same result, whatever it is.”
“So the pattern is the important thing here,” ventured Awkward Goat.
“Exactly,” said Billy Goat Gruff. “It's the pattern. Whether you start with all Red, all Yellow or all Green, you'll get this pattern. And this pattern is associative.”
“Well,” said Awkward Goat, “here are combinations of two Red and a Green, and they all come out Yellow, so this is associative.”
“Do I see another pattern?” wondered Billy Goat Gruff.
“What would that be?” asked Awkward Goat.
“Everything comes out Yellow when you start with a pair of the same color.”
“That's one possibility,” admitted Awkward Goat.
“One possibility!” exclaimed Billy Goat Gruff. “Then what is the other possibility?”
“The other possible pattern,” explained Awkward Goat, “is that the answer is always the third color, the color not in the original line up.”
“I suppose we have to check them, then,” said Billy Goat Gruff.
“It takes some more work,” said Awkward Goat, “but don't you like seeing how it comes out?”
“I think I would like it better,” replied Billy Goat Gruff, “if I were right more often. So two Greens and a Red should be Yellow, then.”
“But it's not,” said Awkward Goat. “We were both wrong. And it's still associative.”
“Green is the identity element,” mused Billy Goat Gruff, “so it cancels itself out. So two Reds and a Green is the same as Red Push Red, which is Yellow. And a Red and two Greens is simply Red, because both Greens cancel out.”
“Well, here are two Reds and a Yellow, and the results are all Red,” said Awkward Goat, “so it's not just the identity element that is ruining our pattern.”
“And here is two Yellows and a Red,” said Awkward Goat, “so it seems our new operation is associative, but it is not as simple as it first appeared.”
“I guess not,” said Billy Goat Gruff.
“Still guessing?” asked Awkward Goat...
Have fun in the comments.