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Space Mystery Magic Trick

January 30th, 2013 by Math Tricks | No Comments | Filed in Math Tricks, Teaching Tricks




A Math Magic Trick

 

This is a great trick just for the sake of performing great tricks.  BUT – it is also a great way for teachers and parents to make algebra interesting to kids!  For all of you teachers and parents out there looking for teaching tricks that will help kids understand algebra, this is one to put into your “teaching tool chest”!  I’ll give you the algebraic explanation of the trick below.  Here, I will tell you about the trick and explain how to do it.

 

You start off with a stack of “Space Mystery” cards (which you can make yourself with our free PDF – the link to that is below), half of which have aliens facing up, and the other have space-only cards facing up.

 

Performing the Space Mystery magic trick is pretty easy once you know how.  You start off with an even number of cards – I would suggest 6 to start off with, and then you can work your way up to whatever number you want, but I would suggest no more than 12.

 

Set up the set of cards such that half of them are showing the alien side, and the other half are showing the space-only side.  Place the cards on the table to demonstrate to your audience that half of the cards are showing the alien side:

      

      

 

Now pick up the cards and shuffle them well:

      

      

 

Place half of the cards on the table in a single row one-by-one:

      

 

With the remaining cards in your hand, use slight of hand to flip them.  I like to switch them into the other hand while I flip them, making it harder for your audience to detect what you did.  Now place the remaining cards on the table in a second row one-by-one.  Magically, the number of cards showing the alien side are the same in each row:

      

As you become more proficient with the trick, you can spiff it up by separating the alien-sided cards from the space-only cards in each row to make the illusion look more striking.

 

Why does this trick work?

How this math magic trick works can be best explained algebraically.  I will use the case where 12 cards are being used.  With a little patience as you go through the explanation, you will soon get the “Ah-Ha!” moment:

You start off with 6 alien cards face-up

 

After the cards are randomized, they are split into 2 stacks of cards, 6 cards each

 

Let the number of alien cards in stack 1 = A

 

Then the number of alien cards in stack 2 must = 6-A

 

Now, with a little thought you can determine the number of space-only cards that remain in stack 2.  This turns out to be 6 minus the # of alien cards, or:

 

6 – (6 – A), which is equal to A

 

So, if you now flip stack 2, the space-only cards (equal to A) now become alien cards – the same number of alien cards in stack 1!

 

 

Printable Cards

 

Here is the free PDF (Space Mystery Printable Cards.pdf) that you can use to print your own cards.  I used the Avery white, two-sided, clean edge business cards (Avery #28878) to print them out.  These can be purchased at any office supply store – I got mine at Walmart!  They came out pretty well – very uniform from card to card.

 

The Alien Storyline

If you are able to perform a dramatic presentation, here is a storyline I came up with that you can use while doing the trick.  You explain to the audience the story behind the aliens:

 

The Talletians are a race of highly intelligent beings from a star system 32 parsecs from our own system.  Throughout their history, they have had an obsession with symmetry.  This obsession has led them to rapid advances in science and technology, which in turn gave them an ability to quickly conquer space.

 

The Talletian obsession with symmetry is evident when they travel through space.  When they are in groups, they prefer to travel such that they are present in equal numbers between groups.  They do not tolerate broken symmetry.

 

You can witness this “need” for symmetry yourself.  The 12 cards in this pack are identical; one side displays a region of space, and the other side displays a Talletian in that space.  Start off by dividing the cards in two groups of six cards each – one with the space side up, and the other with the alien side up.  Now shuffle the cards several times to insure a random distribution.  Now divide the cards into two piles of 6 cards each.  Each pile now has the randomly distributed alien-up cards.  Break this randomness by flipping one pile over.  Now, if you inspect each card in each pile, you will see that each pile contains the same number of alien-up cards.  Symmetry has been made!

 

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Prime Number Tricks

January 3rd, 2013 by Math Tricks | 3 Comments | Filed in Math Tricks, Prime Numbers





One topic that I was introduced to in my elementary school days was the set of prime numbers.  I was given the standard definition of prime numbers that I was to commit to memory, and the composite-prime number lists discussed in class were mostly used as exercises in multiplication and division.  It really is a shame that the more mysterious aspects of prime numbers were not discussed, because it would have perked my interest in mathematics at the time.  I am not a school teacher, so I do not know how prime numbers are taught in schools today.  I can only hope that they are given more attention than when I was in school.  I will not talk about all of the remarkable properties of primes here – I am only going to show a couple of math tricks which are related to prime numbers.

 

Finite or Infinite Primes

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If you are anything like me, you have probably spent many hours contemplating the prime numbers – the set of positive integers (not including 1, as I have to often remind others) that can only be evenly divisible by one or themselves.  A formal definition of a prime number can be stated as:

A number n is prime if it is greater than 1 and has no positive divisors except 1 and n

One of the standard questions one asks about prime numbers is, “How many prime numbers are there?”  The answer is that there are an infinite number of primes.  This was demonstrated by Euclid thousands of years ago.  He came up with a neat trick (described below) to demonstrate how there are an infinite number of primes.

If you take the first few primes and multiply them together, you will come up with a product which is a composite of each prime used, and thus is evenly divisible by each of those primes.  For example:

2 x 3 x 5 = 30

If you divide 30 by 2 or 3 or 5, you will not get a remainder.

Now, if you add 1 to the product, you will get a number that can NOT be evenly divisible by ANY of the primes used.  In fact, this new number can only be either a NEW prime number, or a product of two or more NEW prime numbers.

This is what Euclid did to show how there are an infinite number of primes.  The sequence of results is called the Euclid numbers: 1 + (product of the first n primes).  Applying this method for the first few primes we get:

1 + (2) = 3 (Prime)

1 + (2*3) = 7 (Prime)

1 + (2*3*5) = 31 (Prime)

1 + (2*3*5*7) = 211 (Prime)

1 + (2*3*5*7*11) = 2311 (Prime)

1 + (2*3*5*7*11*13) = 30031 (Not prime, but composed of the new primes 59 × 509)

1 + (2*3*5*7*11*13*17) = 510511 (Not prime, but composed of the new primes 19 × 97 × 277)

1 + (2*3*5*7*11*13*17*19) = 9699691 (Composed of the new primes 347 × 27953)

 

This method can be carried out ad infinitum, demonstrating the infinity of prime numbers.

 

Calculating Prime Numbers

An often-asked question is whether or not you can calculate the number of primes within a stretch of integers.  For instance, how many prime numbers are there from 1 to 3500?  To answer this question, I used Microsoft Access to separate the first 500 prime numbers from the OEIS A000040 (http://oeis.org/A000040) sequence (unverified):

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571

 

So, as you can see, there are nearly 500 prime numbers from 1 to 3500 (489 to be exact), or just about 14%.  Well, that is a lot of work to get the answer you seek.  It would be great if you can at least get an estimate of the number of primes present in a stretch of integers.  As it so happens, prime numbers have been studied for some time now, and there are ways of calculating the number of primes in a stretch of numbers very accurately.  For obtaining a very rough estimate on your calculator, there is a very simple formula you can use that was developed by Carl Friedrich Gauss during his investigations of the relation between logarithms and prime numbers:

Number of Primes from 1 to N ~= N/ln N

So in our example,

# Primes ~ 3500/ln (3500) = 3500/8.16052 = 429 prime numbers

Adrien-Marie Legendre, a contemporary of Gauss, was able to produce a simple formula which produced an even closer approximation:

Number of Primes ~ N/(ln N – 1.08366)

So in our example,

# Primes ~ 3500/(ln (3500) – 1.08366) = 3500/(8.16052 – 1.08366) = 494 prime numbers

I hope this has wetted your appetite for prime numbers.  They really are very interesting, and I am sure you will enjoy reading about them on your own.  Bon Appétit!

 

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