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Today, let's play a game. Playing this game requires a table with unlimited rows and infinite columns, as well as three coins. Three coins are placed in the upper left corner of the table. Just like the illustration below.
The game has only one action. You can remove any coin, but you need to add two coins to the right and bottom of the coin right and bottom. For example, if you remove the second coin above, you need to make up two coins immediately as shown in the picture below.
Let's add another rule. When there is only one coin where the squares on the right and bottom are all empty, you can remove the coin and add two. Even if there is a grid that is not empty, this coin cannot be removed. You can only take other coins that you can take away according to the rules.
Let's circle the grid of 2×2 in the upper left and call it a bank, as shown in the red box section of the figure below.
You will intuitively feel that throughout the game, the coins gradually increase, and the overall flow is to the lower right.
Our question is, what strategy do you have that allows all the coins in the bank to flow out of the bank. That is to say, in the grid of the upper left 2 × 2, there is no longer any coin. Or, you'll feel like there's always going to be coins left in the bank, but you need to state this clearly and precisely.
It's actually a little bit harder, and you can do experiments and think about it for a while. When we solve the puzzle later, you will feel the magic of mathematics. (After a long string of spaces, we start decrypting)
Decryption begins:
You probably did a lot of experiments before getting specific about this. Is it not beginning to doubt that it is impossible to empty the bank if the rules of the game are strictly followed?
That's right! If you have that idea, congratulations, that's the right answer to that question. Well, all that's left to do is to prove your conjecture rigorously.
The method of proof may not be unique, but here is a method that I think is very clever.
Our proof process designs an invariant that we use to illustrate that according to the rules of the game, we can't empty the bank. Invariants are a mathematical concept that, simply put, expresses things that never change in the midst of change, no matter how the situation changes. The idea of invariants runs through almost all branches of the discipline of mathematics, and there is even a view that mathematics itself is a discipline that studies various invariants.
To illustrate the problem, let's do some preparatory work. We mark each grid with a number in the following way.
The rule of the tag is this:
The first row, starting from the first one on the left, is marked 1, 1/2, 1/4,..., in short, the number on the right is half of the previous number.
The second row, marked 1/2, 1/4, 1/8, from the first one on the left,..., is that the starting number is half of the previous row, but always ensure that the number on the right is half of the previous number.
Third row, continue 1/4, 1/8, 1/16,...
And so on...
After the table is marked, you will find that you can take any grid and the number of marks on the right and adjacent grids on its lower side is half of the number of this grid.
Now that we're done, let's design our invariants.
First, let's see how many of the numbers of all the cells would add up. There are infinite sums of numbers, and our strategy is to first find the sum of each line and then sum them up.
The first line is, 1 + 1/2 + 1/4 + ... , the result is 2 ;
In the second line, 1/2 + 1/4 + 1/8 + ... , the result is 1 ;
Third row, 1/4 + 1/8 + 1/16... , the result is 1/2 ;
……
Finally, 2 + 1 + 1/2 + ... , which adds up to 4 .
That is to say, the sum of all the numbers adds up is 4.
Let's look at the rules again. For each operation, we remove the original coin and add two new coins on the right and underside. Although, the coins in the table will be incremented by one, but the sum of the number of tokens in the coin coverage grid on the replacement is equal to the number of coins covered by the previous coin cover marks. This means that no matter how much this operation is, the sum of the number of tokens in the grid covered by all coins remains unchanged.
Well, follow the position of the initial three coins. The sum of the number of marks covered by the three coins is 1 + 1/2 + 1/2 = 2. In a bank, the sum of the token numbers is 2 + 1/4 = 9. To empty all coins inside the bank means that there are no coins in the bank and all coins are outside the bank. According to the previous analysis, it means that to reach a state, the sum of the number of tokens covered by the coins outside the bank remains unchanged from the initial 2. However, the sum of the number of out-of-bank tokens is 4 - 9/4 = 7/4, which is smaller than 2.
So you can't empty the bank.
In addition, in fact, through the previous discussion, you can prove that even if the coin is completely cleared out of the initial three positions, it is impossible. Because, the sum of the remaining token numbers is 2, which requires an infinite number of coins to reach. A finite number of operations, on the other hand, generates up to a limited number of coins.
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