I only have time for a quick post tonight, so let me direct you to one of my favorite math videos. It's of Arthur Benjamin, a mathematician at Harvey Mudd College in California. Art is also a professional magician, and is especially well known for his skill as a lightning calculator. The video is fifteen minutes long, but very enjoyable.
Just so we're clear, his calculations are not tricks. He really is doing what it looks like he's doing. The only portion of the video that could be described as a trick is the part where he determines the missing digit of a seven-digit number after the audience member tells him the other six. That's based on an elementary principle in number theory, which I'm sure some commenter will enjoy explaining.
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I can square 3 digit numbers mentally, but not as fast. I assume it is just a case of learning all the 2 digit squares , and then using (a+b)^2 = a^2 + 2ab + b^2
Hi Steven,
I can square nine-digit numbers mentally, although I very rarely do so because it takes me about 25 minutes, and puts a real strain on my brain. I often find that when I calculate mentally, I crane my head back so that I'm looking up at the ceiling. I don't know why that helps. I also inhale deeply too.
You're right about the squaring trick, but here are a few other tricks. Probably you know some of these. (500+x)^2=((250+x)*1000)+x^2. Also, (500-x)^2=((250-x)*1000)+x^2. That covers numbers in the 400s and 500s. The 900s are fairly easy too: (1000-x)^2=((1000-2x)*1000)+x^2. So are the 100s: (100+x)^2=((100+2x)*100)+x^2. There's a pretty good chance, too, that if you ask someone to "randomly" pick a three-digit number, they'll pick one of the "nice" ones, because humans are hopeless at picking numbers randomly, and they mistakenly imagine that numbers in the 900s are hard to square, when in fact they're really easy. The mathemagician knew that, of course.
It also helps when doing mental arithmetic if you've mentally memorized the double of every integer from 1 to 99, as well as (100-x) for every number from 1 to 99 - which is of course just (99-x)+1. I would also advise memorizing three times each integer from 1 to 99, as well as half of each integer from 1 to 99. That way, you can multiply any number from 1 to 99 by 1,2,3,4,5,6,7,8 or 9, very quickly. For example, 4x47 is double-double 47, and 5x47 is half of 47, times 10. 6x47 is double 3x47, 7x47 is (10x47)-(3x47). 8x47=(10x47)-(2x47). 9x47=(10x47)-47.
Indians watching this video would not be impressed with what the mathemagician did. You see, when they go to school, they learn their multiplication tables up to 100x100. I wish I had done that when I was at school. Maybe I'll get round to it one of these days.
I pity Americans learning their times tables, as they say them the slow way - e.g. "Two times one equals two, two times two equals four," and so on. You'll never memorize them if you learn them like that. When I went to school in Australia, we learnt them like this: "TWO ones are TWO, TWO twos are FOUR, TWO threes are SIX, TWO fours are EIGHT, TWO fives are TEN, TWO sixes are TWELVE, TWO sevens are FOURteen, TWO eights are SIXteen, TWO nines are EIGHteen, TWO tens are TWENty, TWO elevens are twenty-TWO, TWO twelves are twenty-FOUR!" It sounded really exciting, like a train accelerating along a track, and it grabbed the imagination of the boys. We used to have "times tables" races - we'd time ourselves and see how fast we could rattle them off, up to 12x12. That was back in 1970. Sadly, Aussie kids don't learn their times tables now.
Finally, there are a gifted few individuals who really can do sums in their head like Rain Man. They seem to hear a voice in their heads telling them the answer, lucky things! Some of them have synaethesia too. People like that are in a league of their own; they're about 100 times as fast as I am at calculating.
For the trick, it so happens that the number he started with is divisible by 9. Thus he knows that the seven-digit number must be as well, which it is if the sum of its digits is divisible by 9.
I am not sure if he always can avoid making a guess though. Sometimes he might not know whether it is a 0 or a 9 that is the missing digit.
Art told me once how he deals with that situation, but, sadly, I don't remember what he told me. I think he said that he will ask a certain follow-up question which, with a little applied psychology, usually lets him infer whether it was the 0 or the 9 that was omitted.
What would he have done if he hadn't gotten a multiple of 9 from the audience to begin with? Keep going until he got one?
Since he squared four 2-digit numbers from the audience, only one of them has to be a multiple of 3 to result in a multiple of 9, the chance is over 80% in his favour. And I'm sure he has some kind of back-up plan for the other 20% of cases, like simply skipping this part or introducing another trick which gives him more numbers to work it with.
Isn't it almost as easy to cast out 11's as it is 9's, which will help decide if the missing number is a 0 or a 9? I'm guessing here.
Was the number he started divisible by 11 as well?
I enjoyed the video very much. Actually mathematics is an art. It'll be easy considering project-based learning like Arthur have shown us in this video. I enjoyed sitting and learning in our maths tutor classes