In short, put N+ 1 pigeon into n cages, and there must be more than one pigeon in a cage.
Mathematically speaking, if km+ 1 objects are to be put into m boxes, at least one box has at least k+ 1 objects.
Vander Waals's theorem, named after the Dutch mathematician BL van der Waerden, describes:
For the sequence of 1, 2, 3, 4 ... n, if each number is randomly dyed with a color, then there must be k numbers with the same color, forming a arithmetic progression.
As shown in the figure, ***n=8 numbers, and r=2 colors. If we add up the ninth number to be red, then the three red numbers (k=3) of 3, 6 and 9 form a arithmetic progression. If we add up the ninth number to be blue, then the three blue numbers (k=3) of 1, 5 and 9 form the arithmetic progression.
Therefore, according to Vander Waals's numerical calculation, the minimum number of three consecutive arithmetic errors in the case of two colors is nine digits.
Tick-tac-toe is a very simple game with two sides of a circle and a fork. Whoever connects three verticals, three horizontals or three diagonal angles at 45 degrees first wins. As shown in the figure, the fork leans 45 degrees to the right and becomes a line to win.
This chart can be converted to a digital coordinate version:
As can be seen from the above figure, the horizontal 1 1, 12, 13 can win, and the vertical 13, 23, 33 can win. One of the three winning numbers is the same, such as 13, 23, 33.
The winning amounts of diagonal lines are 1 1, 22,33 and13,22,31. The law of this situation is that every number is different. For example, the first number of 13, 22, 3 1 is 1-.
This is the case with two-dimensional coordinates, but it can also be changed into three-dimensional coordinates or four-dimensional coordinates or even more (hypercube).
For this graph, if the second circle crosses the third cross in the first row, it is a draw. However, hales-Jakes theorem points out that when the dimension reaches 8 (that is, 8 numbers are needed for each position), there will be no draw, that is, one side will inevitably be connected into three lines.
The core philosophy of hales-Jeus's special theory is that there is no absolute randomness, and when randomness reaches a certain level, there will inevitably be regular local characteristics.
Local order is a random necessity, and order and randomness are dialectical unity. So life is not the accident of the universe, but the inevitable result of a lot of randomness.
This brings us the following questions:
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