High School Two interesting number theory tricks

[Mark44]

First trick
I learned this one a long time ago and have used it to entertain and amuse young kids. Ask your friend to write down a three-digit number without showing it to you. Then ask him or her to rearrange the digits to form a new three-digit number. After that, write whichever is the larger number above the other number, and then subtract the smaller from the larger, making sure that you don't see any of the numbers.

Then ask the young "victim" to tell you any two of the digits of the answer, with the proviso that if any of the digits is zero, that has to be one of the digits to be reported. I also ask whether the resulting answer has two digits or three.

After he or she tells you two of the digits, you tell the "trickee" the third digit. The key is that in the final answer, the three digits must sum to 9. If, for example, you are told that the digits are 4 and 8, you add the two to get 12, then add them again to get 3. The unknown digit must be 6.

Here's a different example. Suppose your victim writes down 531 and then reverses the digits to form 135. Subtracting the latter from the former results in 531 - 135 = 396. The victim reports that two of the digits are 3 and 6 (and has already stated that none of the digits is 0). A quick mental addition of the reported digits results in 9, so the remaining digit must also be 9.

The number theory behind this trick is modulo 9 arithmetic. Adding the digits of an integer (more than once if necessary) to produce a single digit results in exactly the same value as what you'd get for a remainder if you divided the integer by 9. Reversing or rearranging the digits of this integer produces exactly the same sum. When you subtract one number from another where both are in the same equivalence class, the result will be in the zero equivalence class, modulo 9. IOW, the result will be evenly divisible by 9. It is well known from number theory that numbers that are evenly divisibly by 9 have digits that sum to 9 or an integer multiple of 9.

Second trick
After doing the above, you can continue by asking your friend to reverse the digits of the final answer above, but this time add the two numbers. To make this seem like a real magic trick, ask him or her to tell you any one of the resulting digits. Report back that the remaining digits are whichever of 1, 0, 8, and 9, that you weren't told. I tried this out on my 8-year-old grandson last night, who had some help from his grandmother doing the arithmetic. I had hoped that he would find it interesting, but it turned out that he was completely amazed by it and loved it, going so far as to draw a heart.

Note that the first trick will work with two-digit numbers, three-digit numbers, and so on, and that to get the second number one can rearrange the digits in any order. Presumably the rearranged number is different from the first, or else the result will be zero. The second trick works only on three-digit numbers, and only when the second number is obtained by reversing the digits of the first number.

Here's an example.
First number ----- 842
Second number -- 248
Difference ----- 594

Rev. digits ----- 495
Sum ----- 1089

I've figured out why this works. Can you?

 

[PeroK]

First trick
I learned this one a long time ago and have used it to entertain and amuse young kids. Ask your friend to write down a three-digit number without showing it to you. Then ask him or her to rearrange the digits to form a new three-digit number. After that, write whichever is the larger number above the other number, and then subtract the smaller from the larger, making sure that you don't see any of the numbers.

Then ask the young "victim" to tell you any two of the digits of the answer, with the proviso that if any of the digits is zero, that has to be one of the digits to be reported. I also ask whether the resulting answer has two digits or three.

After he or she tells you two of the digits, you tell the "trickee" the third digit. The key is that in the final answer, the three digits must sum to 9. If, for example, you are told that the digits are 4 and 8, you add the two to get 12, then add them again to get 3. The unknown digit must be 6.

Here's a different example. Suppose your victim writes down 531 and then reverses the digits to form 135. Subtracting the latter from the former results in 531 - 135 = 396. The victim reports that two of the digits are 3 and 6 (and has already stated that none of the digits is 0). A quick mental addition of the reported digits results in 9, so the remaining digit must also be 9.

The number theory behind this trick is modulo 9 arithmetic. Adding the digits of an integer (more than once if necessary) to produce a single digit results in exactly the same value as what you'd get for a remainder if you divided the integer by 9. Reversing or rearranging the digits of this integer produces exactly the same sum. When you subtract one number from another where both are in the same equivalence class, the result will be in the zero equivalence class, modulo 9. IOW, the result will be evenly divisible by 9. It is well known from number theory that numbers that are evenly divisibly by 9 have digits that sum to 9 or an integer multiple of 9.

Second trick
After doing the above, you can continue by asking your friend to reverse the digits of the final answer above, but this time add the two numbers. To make this seem like a real magic trick, ask him or her to tell you any one of the resulting digits. Report back that the remaining digits are whichever of 1, 0, 8, and 9, that you weren't told. I tried this out on my 8-year-old grandson last night, who had some help from his grandmother doing the arithmetic. I had hoped that he would find it interesting, but it turned out that he was completely amazed by it and loved it, going so far as to draw a heart.

Note that the first trick will work with two-digit numbers, three-digit numbers, and so on, and that to get the second number one can rearrange the digits in any order. Presumably the rearranged number is different from the first, or else the result will be zero. The second trick works only on three-digit numbers, and only when the second number is obtained by reversing the digits of the first number.

Here's an example.
First number ----- 842
Second number -- 248
Difference ----- 594

Rev. digits ----- 495
Sum ----- 1089

I've figured out why this works. Can you?

I did this trick once on a climber. I was browsing through his climbing guide book and told him I had memorised all the climbs. All I had memorised was the description of the 9th climb on page 108. So, of course, by this method I got him to pick a climb supposedly at random, then reeled off the name and description of the climb - much to his amazement, I'm pleased to say.

 

[jedishrfu]

I liked a card trick that the YouTube channel Scam Nation streamed. The host is a magician, and a friend stopped by to show the host a new card trick.

The host was blown away by it and asked how he did it because, as a magician, he couldn't figure it out.

The trick was the trick didn't always work when the host heard that he was floored.

The friend continued by saying he would casually tell the audience, "Let's try that again."