April 1 of each year is April Fool's Day in the West, also known as Halloween. On this day, people are allowed to make small jokes with the people around them in various ways, and even if they are fooled, they will not be angry, and they will basically laugh after suddenly realizing. Therefore, on April 1st, there are always all kinds of jokes, but jokes can only be opened until 12 noon at the latest, which is a strict rule.
Image source: google.com
With the continuous strengthening of Sino-foreign exchanges, some Western festivals have also become popular in China, and April Fool's Day is listed among them with its unique humorous characteristics. It is said that on April Fool's Day, many people's mobile phones receive a text message like this: "Please choose a number from 1-9, add 9 minus 8, multiply 9, add ten and one digits, multiply 50 minus 200, please note that this result will represent your character." "After reading this peculiar text message, are you as curious as the owner?" What would be the result?
The next step is what the texter wants: the person who receives the text message is excited to choose a different number to verify it repeatedly. Let's try 7 first: 7+9-8=8,8×9=72,7+2=9,9×50-200=250, huh, surprised, right? Everyone knows that this result is not a good number, because it is a derogatory term that we often use in our colloquial language, and often refers to fools or people who are not serious, who are not serious, who are casual, and who are good at foreign appearances. Isn't it a bit of a cry? If you are not convinced, then try 5 again: 5 + 9-8 = 6, 6× 9 = 54, 5 + 4 = 9, 9× 50-200 = 250, the result is still "holding on to the green mountains and not relaxing". Don't be discouraged, because this trap will "catch everyone". That is, no matter how you choose, no one can jump out of the "fate" of 250.
The saying "two hundred and five" is said to be related to Su Qin, the Zongheng family in the Warring States period
Look at the first step, 1 to 9 to choose a number, add 9 minus 8, equal to this number plus 1, that is, the number we choose must be in 2 to 10; the second step is to multiply these numbers by 9, the results are 18, 27, 36, 45, 54, 63, 72, 81, 90, you must note that this is the key in the game, as long as you observe that the sum of these numbers and the sum of the ten digits is equal to 9, you can realize this; once this step is completed, you will find that the next last step is exactly the same, all 9 × 50-200=250。 Now you've suddenly realized! Text messages are actually constructed using multiples of 9 features, and their mathematical essence is nothing more than 9× 50-200 = 250. The writers just used simple mathematical transformations to quietly tease everyone.
For this kind of mathematical "fool" text message, presumably people will not be grumpy after deciphering it, because at best, it is just a game of fiddling with numbers and playing with mathematics, which is not only harmless, but also fun. However, the following fun problem by the university of Alberta mathematician Leo Mosel for April Fools' Day may take more time and brains to crack, and the mathematical elements contained in it are more worthy of scrutiny and consideration.
The April Fool's Day question is: here's a 28-digit position with 10 vacant positions in between, and if you were asked to fill in the empty slots with the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, each number used only once, what is the probability that the 28 digits you got would be multiples of 396?
5 383 8 2 936 5 8 203 9 3 76
At first glance, the problem is trickier. Because there are 10 × fill in 0 to 9 on the 10 empty slots× 8× 7×6× 5× 4×3×2× 1 = 3628800 cases, the amount of work to verify one by one is obviously unimaginable. So pay attention to the hint of the title, since it is related to April Fool's Day, there must be a mystery in it.
Since it is difficult to operate from a frontal test, it is advisable to do the opposite and start with 396. Because 396= 4×9× 11, a number that is a multiple of 396 must be a multiple of 4, a multiple of 9, and a multiple of 11.
Then, according to the "integral division characteristics of numbers", (1) a number is a multiple of 4, then the last two digits of this number are multiples of 4; (2) a number is a multiple of 9, then the sum of all the numbers of this number is a multiple of 9; (3) a number is a multiple of 11, then the difference between the sum of the numbers on the odd bits of this number and the sum of the numbers on the even bits is a multiple of 11. Then, a multiple of 396 must satisfy all three conditions. With this pertinent conclusion, the answer immediately has a clear direction.
The first step of observation shows that the last two digits of this 28-digit number are 76, which is a multiple of 4, so no matter how you fill in the blanks, you can conclude that the resulting 28 digits must be a multiple of 4; the second step of calculation shows that the sum of all the numbers known in the 28 digits is: 5 +3 +8 +3 + 8 + 2 + 9 + 3 + 3 + 6 + 8 + 2 + 0 + 3 + 9 + 7 + 6 = 90
90 is a multiple of 9, and the sum of the numbers on the vacancy must be 0+1+2+3+4+5+6+7+8+9=45, which is also a multiple of 9, that is, regardless of the order of 0 to 9 on the vacancy, the resulting 28 digits must be a multiple of 9; the third step is still a simple calculation, 5 383 8 2 936 5 8 203 9 3 76 The sum of the numbers on the odd digits is: 5+3+3+8+2+9+6+5+2+3+9+3+73
The sum of the numbers in even bits except the vacancies is 8 + 3 + 0 + 6 = 17, and all ten vacancies appear in the even bits, that is, the sum of the numbers of these ten vacancies is 45 and should also be added to 17, that is, the sum of the numbers on the even bits is 45 + 17 = 62, it can be seen that 73-62 = 11, that is, the difference between the sum of the numbers on the odd bit and the sum of the numbers on the even bit must be a multiple of 11.
Image source: baidu.com
At this point, the truth of the problem has come out. Regardless of how to fill in the 0~9 on the empty bit, all 28 digits obtained are all multiples of 396. They have different forms but similar personalities, just as the so-called turning hands into clouds, overturning hands and raining, but the solution strategy of "not taking the usual road" is worth savoring.
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