A little math magic
“Think of a number” I tell the kids, “and don’t tell me it. Make sure it’s not too big, because you need to do some math with it.”
“Now square your number. You have a result?” I give them a moment, actually after each step. There’s no sense in leaving anyone behind, even the easily distracted. “Take your whole result, and throw it out, except for the rightmost digit, the units digit. Everyone should have a one digit number.”
“Now take that one digit number (everyone has a one digit number?) and square that. Once again, throw it all out, except the rightmost digit. Now, take that last one digit number, and multiply it by your original number.” Fussing with paper, a few calculators… And I ask them one by one what their final result is, and tell them one by one what their original number was.
- “71” “Your number was 71”
- “72” “Your number was 12”
- “75” “Your number was 15”
(more below the fold –>)OK, obviously there is a trick, and they knew that one existed just as well as you do. Still, they played this morning three full rounds (and I got one number wrong, and caught 2 kids who flubbed the arithmetic). And after watching, they hadn’t broken the “code.”
So their assignment (and yours, if you like) : Try this game with a bunch of numbers, starting with 1. I suggested that by 20 or 30 they should know the “rules” to teach someone else to be the mathematician. And I offered a little extra credit. I will report back at the end of the week if anyone did it.
Frankly, I think I could do this with younger kids (these were 9th graders). What do you think?
JD2718 
Very nice problem. Sure gets the kids thinking and makes them feel that math is “fun”.
One question:How do you “guess” their original number when their final answer is 0? For example, what if my original number is 10 or 20 etc?
Good question. The first 0, I guess 10 (today I was correct). The second I guess 20. (Today I was wrong.)
Depending on how many kids try to “solve” this, we might have some fairly interesting discussions. A little modular arithmetic? Divisors of 0?
Perhaps you could start the problem by picking a number that does not end in a 0. Then, for homework have the kids write an explanation,with examples, for why choosing a number that ends in 0 is problematic.
By the way, love your blog. Keep up the good work!
Just a cute application of the little Fermat Theorem, I see. If 5 doesn’t divide x, x^4 ≡ 1 (mod 5). In fact, x^4 ≡ 0, 1, 5, or 6 (mod 10) in any case.
Hm. Looking back, I never did a wrap up post. None of the kids managed to tackle this at home, so one day I wrapped up our lesson fast, divided them into teams, and made them run the trick for every number from 1 through 50. The patterns jumped out, and kids started predicting and testing their predictions.
Then we discussed how to best present their results. Finally, (thanks, Eric) I ‘taught’ them Fermat’s Little Theorem (we had seen the Fermat movie not so long before)
Maybe 3 kids really understood it, but all of them wanted to, and got some idea of what was going on. I’ll keep this in the course.
I’m glad you linked to this one: I’m going to do it. What Fermat movie?
Thanks, I worked on this with some 8th graders in our center, and they enjoyed it, we had a lot of fun with it.
oops, I’m sorry, it was a different problem!
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Ever read the book, Secrets of Mental Math; A Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks by Arthur Benjamin and Michael Shermer? I got it for Xmas and love it. I have always been terrible at math, and feared teaching it, but teaching has actually improved my own skills. This book goes to the next level and makes math fun for me. I’m trying to find more ways of doing these kinds of games with my fifth graders and demystify mental math :)
I’ve seen Benjamin’s stuff, but not the book.
I like the tricks. I like the explanations. I like the feeling of control it gives a kid to learn both the trick and the explanation.
I’m including this in my book, and that made me think: This would be fun to program online. But I don’t know how to program stuff online yet. If I wanted to do that, can anyone here tell me where I would start?
We’re copy editing now. I never did quite understand this trick, and I need to write a footnote explaining it. I just made a spreadsheet to get the results, figured out how the result relates to the number, figured out how to go backward, and am working on understanding why Fermat’s little theorem is true. I’m also working on the relationship between the theorem and the trick. Whew!
Are you good? Let me know if we need a conversation…
Thanks, I worked on this with some 8th graders in our center,