Required Mathematical Intelligence

Chapter 140 Using the Scale to Find the Ball
There are 12 balls, 11 of which are of the same mass and one of which is of different mass.Now how many times would it take to find the ball using a balance without a scale?

[Answer: Divide the 12 balls into 3 groups of 4

The first step is to take two groups out and weigh them. If 4:4 is balanced, the non-standard ones are in the other group of 4.

The second step is to take out 2 balls from that group, and weigh them on the balance with two standard balls. If it is balanced, it will be the remaining 2 balls.

In the third step, the two balls come out with a standard scale.If it is balanced, the non-standard one is the one left, and if it is unbalanced, it is the one on the scale.

Go back to the second step, if it is unbalanced, the non-standard ball is in the two of the upper scales, repeat the third step.Find it from two balls, not standard.

Now discuss the 4:4 unbalanced situation. The remaining 4 balls in the remaining group are all standard. We will use these standard balls for reference later.

The first step, 4:4 unbalanced.

In the second step, take 3 balls from the heavier set and set them aside.Then take 3 from the lighter group and put them in the heavier group.Now the lighter group is left with a ball that may be lighter (not standard) or standard (because it is not known whether the non-standard is lighter or heavier).Take three standard balls and put them on the lighter end.There will be 3 situations, 1, the balance remains the same, 2, balance, 3, the balance is reversed.

The third step is to start with the results of the second step.

1. The result of the second step, if the balance remains the same, it means that the weight of the three heavier balls taken from the lighter ones is the same as the three new standard balls, so the non-standard balls are taken out of the heavier group The remaining one after the three balls and the remaining one after the three balls are taken out of the lighter group, find one of the two balls, and use a standard ball to weigh it.

2. If the balance of the second step is balanced, it means that these 8 balls are all standard, and the non-standard ones are the three balls taken out on one side.Because the three balls are taken out on the heavier side, it can be pushed out that the balls with different qualities are heavier. Find a heavier ball among the three balls, and it will come out in one step.

3. If the height of the balance is reversed, the original lighter end and the remaining one is a standard ball that may be lighter, and now the lighter end becomes heavier, indicating that the remaining one is a standard ball.Similarly, the remaining one at the heavier end is also a standard ball. (Because it used to be heavier, but now it is lighter. If it is not standard, then it is heavier than the standard ball, and the balance will not change on that day.) It means that the non-standard ball gets heavier at the lighter end Among the three balls at one end, because these three balls are at the lighter end, it means that the non-standard balls are lighter than the standard balls. Find a lighter ball among the three balls and just take one step. ]
(End of this chapter)