Required Mathematical Intelligence

Chapter 83 "The 15 Points" Game
The village temple fair has begun.

This year, we played a game called "15 Points".

The entertainer Mr. Carney said: "Come on, folks. The rule is very simple. We just need to put the coins on the numbers 1 to 9 in turn. It doesn't matter whoever puts it first. You put nickels, I put silver dollars, and whoever puts the sum first cover three different numbers of 15, then all the money on the table belongs to him.”

Let's look at the process of the game first: a woman puts it first, and she puts the nickel on the 7, because if the 7 is covered, others cannot put it again.The same goes for some other numbers.

Carney put a silver dollar on the 8.

The woman puts the nickel on the 2 a second time, so she thinks that she can add another nickel to the 6 in the next round to add up to 15, so she thinks she will win.But the entertainer put the silver dollar on the 6 for the second time, blocking the lady's way.Now, he just needs to put the silver dollar on the 1 in the next round to win.

Seeing this threat, the woman puts the nickel on the 1.

Mr. Carney grinned and put the silver on 4 for the next turn.The woman saw that he would win next time on the 5, and had to block his way again, and she put a nickel on the 5.

但是卡尼先生却把银元放在3上，因为8 4 3=15，所以他蠃了。可怜的妇人输掉了这4枚镍币。

Mr. Mayor of the town was fascinated by this game, and he concluded that Mr. Carney had used a secret method so that he could never lose in the game unless he didn't want to win.

The mayor stayed up all night trying to figure out how to do this secret.

Suddenly he jumped off the bed, "Aha! I knew that man had a secret method, and now I know how he did it. Really, there is no way for customers to win."

What trick did the mayor find?You might be able to figure out how to play this "15 point" game with your friends without losing a game.

[Answer: The trick to understanding the game of 15 is to see that it is mathematically equivalent to the game of Tic Tac!Surprisingly, the equivalence relationship is established on the basis of the famous 3×3 Rubik’s Cube, which was discovered in ancient China.

要了解这种魔方的妙处，先列出其和均等于15的所有三个数字的组合（不能使两个数字相同，不能有零）。这样的组合只有八组：1 5 9=15；1 6 8=15；2 4 9=15；2 5 8=15；2 6 7=15；3 4 8=15；3 5 7=15；4 5 6=15.
Now let's take a closer look at the following unique 3×3 Rubik's Cube
It should be noted that there are eight groups of elements, all on eight straight lines: three rows, three columns, and two main diagonals.Each line is equivalent to one of eight sets of three numbers (which add up to 15).Therefore, in the competition game, each group of winning three numbers is represented on the square matrix by a certain row, a certain column or a certain diagonal line.

Obviously, each game has the same rationale as the "tic-tac-toe" game played on the square.Mr. Carney, the entertainer, drew a magic quadrant on a card and placed it under the playing table, where only he could see it (no one else could see it).There is only one position of the magic quadrant structure, but it can be rotated to produce four different combined forms, and each form can produce another four forms through reflection, and there are eight forms in total.Each of these eight forms can be used as a recipe when playing this game, all with the same effect.

While playing the "15 points" game, the artist Mr. Carney secretly played the corresponding "tic" game on the card drawing.There is no losing in this game, and if both sides play correctly, there will be a draw in the end.However, people who participate in entertainment competitions are always at a disadvantage because they have not mastered the secret of "tic word game".Mr. Carney the Entertainer could therefore easily set up an ambush that would surely win. ]
(End of this chapter)