What is the maximum number In 2048?

Michael brand raised the following question in the April 2014 puzzle of using your head is permitted quenti: In the recently popular mini game 2048, what is the maximum number you can get?

Here, we will briefly describe the rules of the game. The game is played on a 4x4 Board. The Board is filled with "several" blocks, each of which has a positive integer in the form of 2n. In each step, you need to select a direction from the top, bottom, left, and right directions. After you press the corresponding direction key, all the blocks will "fall" into this direction; if two of the same types of blocks are collided during this process, their values are added and a new number is merged. Then, the system randomly selects a blank position in the checker board and generates a new number of blocks with numbers 2 or 4 (9: 1 in both cases ). At the beginning of the game, two random blocks will be automatically generated on the board. Your goal is to get a number with 2048 pieces written through limited steps. Of course, even if you get 2048, the game will not end automatically, and you can challenge a larger number. So now we have the question: theoretically, what is the maximum number of players in this game?

It can be proved that we can never play the number 2048 in 218.

Let's add up all the numbers on the board and pay attention to the Binary Expression of the current sum during the accumulation process. If the number in the board is 2, 4, 16, 64, 16, 2, the Binary Expression of the accumulated result is 10 → 110 → 10110 → 1010110 → 1100110 → 1101000. You will find that since each number on the board is a positive integer in the form of 2n, after you add it to the sum, the Binary Expression of the sum will only have moreOneNumber 1 (if carry occurs, the number of Number 1 may not change or even decrease ). This means that if the Binary Expression of the sum of all numbers on the final board contains k numbers 1, it means that at least K numbers are added, in other words, there must be at least K blocks in the board.

It is easy to see that after each step, the sum of all numbers on the board will increase by 2 or 4. If a 218 error occurs on the board, it means that the sum of all the numbers on the board is at least 218, before that, the sum of all the numbers on the board must have gone through 218-2 or 218-4. The Binary Expression of the former is 11 1111 1111 1111 1110. There are 17 digits 1, which exceeds the total number of grids in the board. Therefore, it is obviously impossible. The Binary Expression of the latter is 11 1111 1111 1111 1100, where there are 16 digits 1, which is the total number of cells in the board. This shows that the board is filled up, 22, 23, 24 ,..., 217 each of the 16 different data blocks has one. This means that there are no blank grids in the Board, and there are no numbers of the same types that can be merged. At this time, the players will die! Therefore, we cannot get 218.

Therefore, 217 = 131072 is a theoretical ceiling on the size of several blocks in the 2048 game. But can we get 131072? It seems that we only need to construct the shown situation. However, whether the situation can be achieved remains to be further discussed. Michael brand guessed that 131072 could not be obtained theoretically, but he could not prove it. Although I have seen many online users claim that they have played 131072, some even sent the last few videos (such as here), but since I have never seen a complete demonstration process, therefore, they are skeptical. We hope that our omnipotent netizens can provide a concise and effective constructor to show that when we are lucky enough, we can beat 131072 or find a better way to prove it, it is enough to prove that 131072 is theoretically impossible.