Prepare for the "worst" first. Fly into each nest 1 pigeon. No matter which nest the remaining pigeons fly into, there are always 1 nest with at least 2 pigeons.
If there are three drawers, what would happen if mom bought four apples and asked you to put them in the three drawers?
We can first divide 4 into the sum of several integers, and there are four situations as follows:
4=4+0+0
4=3+ 1+0
4=2+2+0
4=2+ 1+ 1
Observing the above four ways of placing apples, we find a similar property: no matter which way of placing apples, there are always two or more apples in a drawer. In other words, if you put four apples in three drawers, there will always be at least two apples in one drawer.
If you increase the number of apples, put five apples in four drawers. No matter which way you put them, there must be at least two apples in one drawer. This is the pigeon coop principle:
There are m items. Put them in n drawers. If there are more items than drawers (that is, m is greater than n), there must be more than two items in a drawer.
Example 1: Three children walk together, and two of them must be of the same sex.
Analysis: There are only two sexes: male and female. We regard sex as two drawers and compare three children to apples. The number of apples is three more than that of drawers. According to the pigeon hole principle, at least one "drawer" contains two or more "apples", which means at least one child is of the same sex.
Example 2: Li Shifu is repairing a machine. There are four pairs of nuts in red, yellow, blue and white in the toolbox, but the light bulb in the room suddenly broke down, so Li Shifu had to take the nuts out of the room to identify them. How many nuts does Li Shifu need to eat to ensure that a pair of nuts are the same color?
Analysis:
If Li Shifu only takes two nuts, can he guarantee the same color?
(2) If you start with red and yellow, can you guarantee that only one pair of colors are the same? How about two more glasses? Why?
(3) Take at least a few, can you ensure that there are two nuts with the same color?
(4) If nuts are red, yellow, blue, white and black, at least a few nuts should be taken to ensure that a pair is of the same color? Did you find a pattern?
Solution: Li Shifu must take at least five nuts to make sure that one pair is the same color.
Example 3: There are four kinds of glass balls with different colors in the pocket. Pull them out two at a time. How many times do you have to touch 10 to ensure the same result?
Analysis: When two balls are the same color, there are four different results. When the colors of two contacted balls are different, there can be at most 3+2+ 1 = 6 different results. Take 4+6 = 10 (kinds) different results as drawers.
Solution: Because it takes 10 times to get the same result, according to pigeonhole principle, at least touch 9× 10+ 1 = 9 1 (times).
Example 4: A box contains 10 red, yellow and blue jellies. How much jelly do you have to eat to ensure that there are at least two pairs of jelly with different colors?
Analysis: Make sure that at least two pairs of jellies have different colors. From the most unfavorable situation, first take 10 jellies with the same color, and the remaining two colors can be regarded as two drawers, and the result can be obtained.
Solution: If you take 10 jellies with the same color, then the remaining two jellies can be regarded as two drawers, which is more than the number of drawers 1, that is, taking three jellies will definitely get another pair of jellies with the same color. In this way, at least 13 jellies can be taken, and at least two pairs of jellies with different colors can be guaranteed.
Example 5: There are some balls of different colors in a carton, including 10 yellow balls, 9 white balls, 8 black balls and 2 purple balls. How many balls does Xiao Ming take out with his eyes closed to ensure that at least four balls have the same color?
Analysis: To take out four balls with the same color, they can only be yellow, white and black, not purple balls, because there are only two purple balls. Suppose you are unlucky and get two purple balls, then there are only three colors left: yellow, white and black. Think of these three colors as three drawers.
Solution: Suppose that two purple balls have been taken, and the remaining three kinds of balls are regarded as three drawers, and three balls are put in each drawer, then take 3× 3 = 9 (balls). If you take one more ball, you can ensure that the colors of the four balls are the same. That is, 2+9+ 1 = 12 (balls) can ensure that the colors of the four balls are the same.
Exodus 6: How many cards must be taken out of a deck of playing cards to ensure that the cards taken out have four colors?
Analysis: If the king of size is drawn at the beginning, thirteen hearts are drawn at the back, thirteen spades are drawn at the back, and thirteen diamonds are drawn at the back, then it is 2+ 13 × 3 = 4 1+1= 42 (cards).
Solution: 2+ 13× 3+ 1 = 42 (Zhang)