The following operations are performed on the natural number: if it is even, divide it by 2; if it is odd, subtract 1 until the result becomes 0. Then after four operations, there are ▁ numbers that make the result become 0, which are ▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁

The following operations are performed on the natural number: if it is even, divide it by 2; if it is odd, subtract 1 until the result becomes 0. Then after four operations, there are ▁ numbers that make the result become 0, which are ▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁

8、0
8\2=4 4\2=2 2\2=1 1-1=0
0 / 2 = 0 (the same as above)

Will 1, 2, 3 These 100 natural numbers are randomly divided into 50 groups with two numbers in each group. Mark one number as a and the other number as B, and substitute it into algebraic formula 12 (a + B − | a − B |) to calculate the result. After 50 groups are substituted, 50 values can be obtained, and the minimum sum of these 50 values can be obtained (please explain the reason briefly)

The reason is as follows: suppose a > b, then 12 (a + B - | A-B |) = 12 (a + B-A + b) = B, so when the smaller number B in the group of 50 is exactly 1 to 50, the sum of these 50 values is the smallest, and the minimum is 1 + 2 + 3 + +50=50(1+50)2=1275．

Put 1, 2, 3 The 100 natural numbers are randomly divided into 50 groups with two numbers in each group. Now, mark any number in each group as a and the other as B, and substitute it into the algebraic formula 0.5 (| A-B | + A + b) to calculate the result. After 50 groups of numbers are substituted, 50 values can be obtained, and the maximum sum of these 50 values can be obtained,

① If a ≥ B, the absolute value sign in the algebraic expression can be removed directly,
The algebraic expression is equal to a,
② If b > A, the sign of absolute value is opposite,
The algebraic expression is equal to B
It can be seen that by inputting a pair of numbers, you can get the largest number in the pair (it has nothing to do with who is a and who is b)
Since it's a summation, it's necessary to add up these 50 numbers to the maximum,
We can enumerate a few groups of numbers to find the law,
If 100 is combined with 99, 99 is wasted,
Because if you enter 100 and 99, you only get 100,
If we take two sets of numbers 100 and 1, 99 and 2,
The sum of these two groups of numbers is 199,
If we take 100 and 99, 2 and 1,
Then the sum of these two groups of numbers is 102,
In this way, it is obvious that the meeting of big numbers and big numbers should be avoided, so that the last and the largest numbers can be made. Therefore, as long as the largest 50 numbers in the 100 natural numbers are in different groups from 51 to 100,
In this way, the sum of 50 numbers is finally obtained, and the maximum value is the sum of 50 numbers from 51 to 100,
51+52+53+… +100=3775．