How to square any number between 1 and 100

In my post on magic mathematical tricks I claimed to be able to easily square any number between 1-100. Later I got a comment enquiring about this claim, according to my reader there is only a calculation "fast track" for numbers ending with 5, i.e. 5, 15, 25, 35, 45, 55 etc, and not for all the other numbers between 1-100.

The fact of the matter is that numbers ending with 5 are particularly easy to square, but there is a more general rule which works for all numbers. Here is how you do it…

Take any number between 1-100… 67, ok, lets square 67. First you find a number close to 67 that is easy to multiply such as 70. Next you move the same number of steps (there are 3 steps from 67 to 70) in the opposite direction giving you 64. Multiply 70 and 64 (4200 + 280= 4480). Then add the square of the number of steps that you took in either direction i.e. 3, giving you 9. Now simply add these two numbers to get 4489, which is the square of 67.

Ok sure, multiplying 64 and 70 does require some mathematical skull but is not that hard and most people can do it with a little bit of training I think. The trick is much easier for smaller numbers such as 18. For 18 you would multiply 20 and 16, giving you 320, plus the square of 2, which is 4, giving you a total of 324. Here is an illustration of the method taken from the teaching company course "The joy of mathematics".

The very easiest cases are numbers ending with 5. Take the number 45 and apply the previous rule. You go five steps in either direction giving you 40 and 50. Multiplying these gives you 2000. Add the square of the number of steps that you took i.e. 5, giving you 25. So the square of 45 is 2000 + 25 = 2025. For numbers ending with five you can simply multiply the number below and above the first number and then write 25 behind that number i.e. 4*5 = 20, and with 25 behind that gives you 2025. For 65 it would be 6*7 = 42 and then 25 behind that giving you 4225, for 15 it would be 1*2 = 2, and then 25 behind giving you 225.

This trick seems to work well with higher numbers as well. The only limitation is your multiplication skills. So go out and impress your friends. Next time I write about math I will explain how you can pull of the almost autistic feat of telling what day of the week it was or will be at any date in history or future

Magical mathematical tricks

Parallell to working on forthcoming articles I have been listening to the Teaching Company series called "The joy of mathematics", taught by professor Arthur T. Benjamin (see picture).

It is a truly great series of lectures which has allowed me to fresh up my math as well as to learn som impressive new skills. For example I can now square any number between 1-100 very fast (perhaps I will reveal the trick in a subsequent post). The lecturer has also perfomed two "magical tricks". The first one I think most people will be familiar with, I recall hearing it in 3rd grade, it goes as follows:

1. Think of a number between 1-10
2. Double that number
3. Add 10
4. Divide that number by 2
5. Subtract the number that you first thought about

Ok, now concentrate on the number that you are left with, a little bit more... I think I am getting... wait for it... yes, I almost have it... 5! Was that the number that you were left with? If not, it is because you did an error. Some simple algebra will prove why you are always left with 5.

x*2 = 2x,
2x+10 = 2x+10,
(2x+10)/2 = x+5,
x+5-x = 5

If you want to end up with a different number you can merely swap out the 10 with something else. Your answer will always be half of the number that is added in the second step.

The second magical trick is somewhat more complicated and also more impressive if you ask me. It goes as follows:

1. Think of a number between 1-10
2. Triple that number
3. Add 6
4. Triple the number again
5. Now add the individual digits in your numer
6. If you still have a two digit number, add the individual digits again

Now, concentrate again. Unless you have done a mistake you are thinking about the number 9. The following algebra plus some explanation shows why this trick works.

x*3 = 3x
3x+6 = 3x + 6
(3x+6)*3= 9x+18

Now 18 is a multiple of 9 and adding x number of nines to that will always result in a multiple of 9. Multiples of nine always consists of digits that add up to 9, or to another multiple of 9 which will then add up to 9. You can check this out for yourself. The multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126 etc

Adding the individual digits of any of these number will give 9 except 99, which gives 18 which you then add again and get 9.

James Randi

This week I want to pay a tribute to a man that I admire very much, the skeptic James Randi (see picture) who will turn 81 in august. Randi, in a sense, is a man with two faces. First there is "The Amazing Randi", a world famous magician who has performed many seemingly impossible tricks. Among his achievements Randi is the holder of two world records. On one occasion Randi outperformed the infamous Harry Houdini when he spent 104 minutes in a sealed coffin on the bottom of a swimming pool. He earned his second entry into Guiness world records when he came out alive after having been frozen in a block of ice for 55 minutes. Does anyone know how this was done?

Magicians are of course interesting and fun, however, it is not magic skills that makes Randi unique. Like the just mentioned Houdini, James Randi is a skeptic. He has openly admitted that he has never performed any real magic in the supernatural sense. In other words, all the magical shows that we laymen find impossible or at least incomprehensible have merely been trickery. This of course begs the question of whether other people who seem to have similar magical talents are also just using tricks. My belief is that they are merely using tricks, and I have never encountered any evidence (excluding anecdotal stories) that has made me contemplate this. Interestingly Randi has once been accused of being a fraud in the opposite sense. Once when he performed a man claimed that Randi was in fact using psychic powers and that he is lying when he claims he is using trickery.

Nevertheless, our dear magician is perhaps most famous for his one million dollar paranormal challenge. James Randi has 1.000.000 dollars waiting in a bank account for anyone who can demonstrate that he/she has psychic/supernatural powers. Hundreds have tried but no one has succeeded until this day. Others, including the previously mentioned Houdini, and the Swedish organization Humanisterna have posed similar challenges. (Just recently, a psychic in Sweden failed a mutually agreed upon test, getting zero correct of a possible fifteen.)

Through the years James Randi has appeared on TV quite a few times. If you go to "You Tube" and search for James Randi you will find many interesting clips to watch. Alternatively, you can also go to his website, http://www.randi.org/, where there is a nice collection of multimedia as well as many interesting articles.

Below I have put together my own list of my favorite Randi multimedia including some personal comments. Enjoy!

Uri Geller and Peter Popoff. In this video Randi explains what he is doing and why he is doing it. Randi also shows how he exposed Uri Geller (see picture), the spoon bender phenomenon for what he really is, a fake (or perhaps he really just had a bad day). Finally he shows how Peter Popoff, a priest who makes miracles happen, was actually just using a wireless radio (Popoff went bankrupt a year after he had been exposed).

James Hydrick. James Hydrick, a 23 year old martial arts trained psychic, claims that he can turn pages the pages of a phone directory using his mental powers (and that we all can learn to do it). By using a simple control Randi shows that Hydrick is not using any psychic powers, but rather he is simply blowing. According to wikipedia, Hydrick has subsequently admitted that he was indeed a fraud, what a surprise…

Astrology. An entire class is given identical horoscopes and are asked to rate the accuracy. Everyone in the class rates the accuracy as being either four or five out of five.

Psychic surgery. In this video Randi performs, using only his bare hands, what looks like a very real surgery in order to show that no supernatural powers must be involved (as some others have suggested)

Homeopathy. This video is actually one of my favorites. Randi has a 15min lecture in which he explain the very odd principles of homeopathy. He also tells us about when he downed no less 64 tablets without being affected at all. This is despite the fact that that the cans say that you should call a poison doctor if you take 4 or so (I cannot remember the exact numbers).

Spoon bending. Randi shows us how he can bend a spoon using trickery. It looks just like when Uri Geller bends his spoons using his psychic powers, or perhaps Geller was also using trickery? You decide…

Medium discussion, Part 1, Part 2, Part 3. Randi is on Larry King live where he discusses an incident where the medium Sylvia Browne told the parents of an abducted child that their child was dead. The women who got the erroneous information from her spirits still claims to have psychic powers, she have accepted Randi challenge, but have still not actually done it. Afraid that she won't make it perhaps?

James Randi and a medium. In this video Randi exposes a medium by revealing a transcript in which the medium has given her client no less than 37 names, and asked him whether he feels there is any connection to any of these names. Statistically speaking it would be quite unlikely if the client had no connection to any of the names...

Applied Kinesiology. In this video James Randi, and a kinesiologist do a blind study, testing whether a particular chosen crystal make the test person stronger. If you are to believe the result in this particular test, the subject benefitted very much from rat poison, hmmm...

If anyone has any further multimedia tips I will watch them with much interest.