Mathematics Fun, Fact, Fiction, Function, Fantasy 
Mathematics Fun, Fact, Fiction, Function, Fantasy
Brand new 2002 Lifesmith
line of imagery --- The P Line ---> Take
me there!Here's a fun trick to show a friend, a group, or an entire class of people. I have used this fun, mathematical trick on thousands of people since 1963 when I learned it. Tell the person to think of his/her birthday; that you are going to guess it.
Step 1) Have them take the month number
from their birthday: January = 1, Feb = 2 etc.
Step 2) Multiply that by 5
Step 3) Then add 6
Step 4) Then multiply that total by four
Step 5) Then add 9
Step 6) Then multiply this total by 5 once again
Step 7) Finally, have them add to that total the day they were
born on. If they were born on the 18th, they add 18, etc.
Have them give you the total. In your head, subtract 165, and you will have the month and day they were born on!
How It Works: Let M be the month number and D will be the day number. After the seven steps the expression for their calculation is:
5 (4 (5 M + 6 ) + 9 ) + D = 100 M + D + 165
Thus if you subtract of the 165, what will remain will be the month in hundreds plus the day!
2) Divisibility Rules! To find if a number X is divisible by a certain number, test the number by using the information in the table below.
| By 2 | If the last digit divisible by two, then X is too |
| By 3 | If the sum of the digits of the number X is divisible by three, then X is too |
| By 4 | If the last two digits are divisible by four, then X is too |
| By 5 | If the last digit is 5 or 0, then X is divisible by 5 |
| By 6 | If X is divisible by 2 and by 3, then X is divisible by 6 |
| By 7 | This rule is called L-2M. What you do is to double the last digit of the number X and subtract it from X without its last digit. For instance, if the number X you are testing is345678, you would subtract 16 from 34567. Repeat this procedure until you get a number that you know for sure is or is not divisible by seven. Then the X's divisibility will be the same. |
| By 8 | If the last three digits are divisible by 8, then X is too |
| By 9 | If the sum of the digits of the number X is divisible by nine, then X is too |
| By 10 | If the last digit of X is 0, then X is divisible by 10 |
| By 11 | What you do here is to make two sums of digits and subtract them. The first sum is the sum of the first, third, fifth, seventh, etc. digits and the other sum is the sum of the second, fourth, sixth, eighth, etc. digits. If when you subtract the sums from each other, the difference is divisible by 11, then the number X is too |
| By 12 | If X is divisible by 4 and by 3, then X is divisible by 12 |
| By 13 | This rule is called L+4M. What you do is to quadruple the last digit of the number X and add it from X without its last digit. For instance, if the number X you are testing is345678, you would add 32 to 34567. Repeat this procedure until you get a number that you know for sure is or is not divisible by thirteen. Then the X's divisibility will be the same. |
| By 14 | If X is divisible by 7 and by 2, then X is divisible by 14 |
| By 15 | If X is divisible by 5 and by 3, then X is divisible by 15 |
| By 16 | If the last four digits are divisible by 16, then X is too |
| By 17* | This rule is called L-5M. See rules for 7 and 13 on how to apply. |
| By 18 | If X is divisible by 9 and by 2, then X is divisible by 18 |
| By 19* | This rule is called L+2M. See rules for 7 and 13 on how to apply. |
| By 20 | If X is divisible by 5 and by 4, then X is divisible by 20 |
| By 21 | If X is divisible by 7 and by 3, then X is divisible by 21 |
| By 22 | If X is divisible by 11 and by 2, then X is divisible by 22 |
| By 24 | If X is divisible by 8 and by 3, then X is divisible by 24 |
| By 25 | If the last two digits of X are divisible by 25, then X is too |
*A big thank you to Torsten Sillke for these rules!
3) Alphametics --- the alphabetic and mathematical construction where each letter is represented by a unique number in the problem. Each of these has a unique solution.
| SEND+MORE=MONEY | FIFTY+STATES=AMERICA |
| EARTH+AIR+FIRE+WATER=NATURE | TERRIBLE+NUMBER=THIRTEEN |
| SATURN+URANUS+NEPTUNE+PLUTO=PLANETS | GEORGIA+OREGON+VERMONT=VIRGINIA |
| SIX+SIX+SIX+BEAST=SATAN |
4) Good Mathematical Card Trick
Here's a real clean card trick that is bound to amaze and surprise your friends.
Have a friend shuffle a standard 52-card deck to his satisfaction. The ask him/her to turn over, face up, a pile of twenty-five cards. As they count out the cards to twenty-five, act like you are intensely memorizing every single one of them in order. In truth you are only interested in the 17th card in the pile. Memorize that 17th card..
Turn the twenty-five card pile over, now face down, and set aside.
Take the remaining cards and do the following:
If it is a two through nine card, place it face up, and count out cards up to the number ten. For instance, if you turn up a six, you would place it face up, and then count out four cards coming down, saying "Seven, eight, nine, ten." If you turned up a three, then seven cards, saying "Four, five, six, seven, eight, nine, ten."
If the card you turn up is an ace, ten, or picture, tell your friend that you must have cards 2 through nine for this trick and place the card on the bottom of the deck, face down like the rest of the cards. (Someone told me that aces and pictures work too, but I haven't verified it yet.)
Repeat this procedure until you have completed four columns of "ten counts."
Take whatever remaining cards you have after making the four columns, and place them face down on TOP of the twenty-five card pile you set aside earlier.
Now ask your friend to total up the numbers at the top of the four columns.
Say the total was 23. Counting from the very top of the set-aside-pile-plus-remainder-cards, state you are interested in the 23rd card. Just before turning over the 23rd card, state that it is the 17th card that you memorized at the beginning of the trick!
If you did everything correctly, it works every time! Amazing!
Did you know that 27^5 + 84^5 + 110^5 + 133^5 = 144^5 ?
Did you know that 9^3 + 10^3 = 12^3 + 1^3 ?
Did you know that 2682440^4 + 15365639^4 + 18796760^4 = 20615673^4 ?
Did you know that 95800^4 + 217519^4 + 414560^4 = 422481^4 ?
Did you know that 111,111,111 x 111,111,111 = 12,345,678,987,654,321 ?
Did you know that 123,456,789 x 989,010,989 = 122,100,120,987,654,321 ?
Did you know that e^(pi * sqrt(163)) = 262537412640768743.9999999999992 ? Coincidence? Try Here!
Did you know there are just four numbers (after 1) which are the sums of the cubes of their digits: 153 = 1^3 + 5^3 + 3^3, 370 = 3^3 + 7^3 + 0^3, 371 = 3^3 + 7^3 + 1^3, and 407 = 4^3 + 0^3 + 7^3 ?
Did you know 1,741,725 = 1^7 + 7^7 + 4^7 + 1^7 + 7^7 + 2^7 + 5^7 ?
6) Trachtenberg System of Speed Mathematics
When I was just seven or eight years old, I came upon a most fascinating book called the Trachtenberg System of Speed Mathematics (or something like that). In it was described a way of doing addition, subtraction, multiplication and division in ways I had never seen before. None of my teachers had even heard of it before. I was able to do multiplications by eleven and twelve in my head faster than my friends could do on paper. If you hate formal math (or if you have a child that does), I urge you to download the software that lays it all out for you.
If the link is down, email me and I will send it along to you. Another site which offers it is here.
Jakow Trachtenberg was born on June 17, 1888, in Odessa, Russia. He worked as an engineer during his younger years in the Obuschoff Shipyards, after graduating with highest honors from the Mining Engineering Institute in St. Petersburg. He moved to Berlin during World War I and soon became an expert in Russian affairs. There he devised a method of teaching languages that is still used in some parts of eastern Europe. As a Jew, he was captured by the Nazis and deported to a concentration camp. There, without paper or pencil (some say a common iron nail became his most prized possession), he devised a method of doing mental arithmetic. Trachtenberg managed to escape and fled to Switzerland, where he perfected his system. In 1950, he founded the Mathematical Institute in Zurich where he taught his methods to children and adults alike.
One source of info about the system / Download the software here.
7) Facts about Regular Polygons and Polyhedra
The following legend pertains to the table below it:
| s = length of a side | A = area | d = diagonal |
| R = radius of circumscribed circle or sphere | V = volume | a = length of an edge |
| r = radius of inscribed circle | h = height | S = surface area |
| Triangle | A = ((s^2)*sqrt(3)) / 4 | Tetrahedron | V = (1/12) * sqrt(2) * a^3 |
| R = (1/3) * s * sqrt(3) | S = a^3 * sqrt(3) | ||
| r = (1/6) * s * sqrt(3) | R = (1/4) * a * sqrt(6) | ||
| r = (1/12) * a * sqrt(6) | |||
| Square | A = s^2 | Cube | V = a^3 |
| R = (1/2) * s * sqrt(2) | S = 6 * a^2 | ||
| r = (1/2) * s | R = (1/2) * a * sqrt(3) | ||
| r = (1/2) * a | |||
| Pentagon | A = (s^2 / 4) ( sqrt( 25 + 10 * sqrt(5) ) ) | Octahedron | V = (1/3) * a^3 * sqrt(2) |
| R = (1/10) * s * sqrt (50 + 10 * sqrt(5) ) | S = 2 * a^2 * sqrt(3) | ||
| r = (1/10) * s * sqrt (25 + 10 * sqrt(5) ) | R = (1/2) * a * sqrt(2) | ||
| r = (1/6) * a * sqrt(6) | |||
| Hexagon | A = (3/2) * s^2 * sqrt(3) | Dodecahedron | V = (1/4) * a^3 * (15 + 7 * sqrt(5) ) |
| R = s | S = 3 * a^2 * sqrt(25 + 10 * sqrt(5) ) | ||
| r = (1/2) * s * sqrt(3) | R = (1/4) * a * ( sqrt(3) + sqrt(15) ) | ||
| r = (1/4) * a * sqrt( (50 + 22 * sqrt(5)) / 5 ) | |||
| Octagon | A = 2 * s^2 * (1 + sqrt(2) ) | Icosahedron | V = (5/12) * a^3 * (3 + sqrt (5) ) |
| R = (1/2) * a * sqrt( 4 + 2 * sqrt(2) ) | S = 5 * a^2 * sqrt(3) | ||
| r = (1/2) * a * (1 + sqrt(2) ) | R = (1/4) * a * sqrt(10 + 2 * sqrt(5) ) | ||
| r = (1/2) * a * sqrt( (7 + 3 * sqrt (5)) / 6 ) | |||
| Decagon | A = (5/2) * s^2 * sqrt (5 + 2 * sqrt(5)) | Sphere | V = (4/3) * pi * a^3 |
| R = (1/2) * a * (1 + sqrt(5) ) | S = 4 * pi * a^2 | ||
| r = (1/2) * a * sqrt( 5 + 2 * sqrt(5) ) | |||
8) Magic Trick --- Take any whole number greater than 0. Take half of it if it is even or triple it and add one if it is odd. Repeat over and over again. You'll always get the same result ---> 1.
9) The Five Most Important Numbers in Mathematics in One Equation
e^i*pi + 1 = 0
By the way, you can see the value of e to 10,000 digits Here!
10) Infinite Integer Sequences? There's a whole encyclopedia devoted to them! HERE!
11) Primes! As of 1998, the largest known prime number is 2^(3021377) - 1 . Known as the 37th Mersenne prime number, that is, of the form 2^p - 1, where p is also prime, it has more than 909,000 digits! You can seeit Here! For more info on Mersenne primes, go HERE! - WAIT!!- NEWS FLASH - An even larger Mersenne prime has been found! Yes, believe it or not, the 39th Mersenne prime has been verified as 2^13466917 - 1, an astounding number having 4,053,946 whopping digits...there is a poster available of it too...I'll have to find where...
12) What a Slice of Pi! As of 1998, the greatest calculation of pi has been done by two Japanese researchers at the University of Tokyo to have 51,539,600,000 digits! That's over 51 billion digits! There is an interesting report on it plus lots of curious facts about it too...go Here! For the first 100,000 digits of pi, go HERE! You can even search for your birthday in pi! Try Here.
13) Here is a 160 character program in C, written by D. T. Winter, which will calculate the first 800 digits of pi:
int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5;
for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a,
f[b]=d%--g,d/=g--,--b;d*=b);}
14) Strange Properties of 666, "The Number of the Beast"
The last book of the Bible, Revelations, brings up the number 666 as being the number of the beast connected with the end of this age and the coming of the Messiah. You will find the direct reference in Chapter 13, verse 18 of Revelations. Besides that cataclysmic reference, the number 666 has quite a few very interesting properties.
666 = 3^6 - 2^6 + 1^6
666 = 6^3 + 6^3 + 6^3 + 6 + 6
+ 6
(Mike Keith mentions that there are only five other positive integers
that exhibit this property...find 'em!)
666 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2
666 = 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9 = 9 + 87 + 6 + 543 + 21
Moreover, 666 is equal to the sum of the cubes of the digits in its square (666^2 = 443556, and the sum of the cubes of these digits is 4^3 + 4^3 + 3^3 + 5^3 + 5^3 + 6^3 = 621) plus the sum of the digits in its cube (666^3 = 295408296, and 2+9+5+4+0+8+2+9+6 = 45, and 621+45 = 666).
Incredibly, The number 666 is equal to the sum of the digits of its 47th power, and is also equal to the sum of the digits of its 51st power. That is,
666^47 = 5049969684420796753173148798405564772941516295265
4081881176326689365404466160330686530288898927188
59670297563286219594665904733945856
666^51 = 9935407575913859403342635113412959807238586374694
3100899712069131346071328296758253023455821491848
0960748972838900637634215694097683599029436416
and the sum of the digits on the right hand side is, in both cases,
666. In fact, 666
is the only integer greater than one with this property.
(Also, note that from the two powers, 47 and 51, we get (4+7)(5+1)
= 66.)
Mr. Keith also points out that if we assign numerical values for the letters of the alphabet starting with A = 36, B = 37, and so on, we find that the letters in the word
SUPERSTITIOUS = 666 !!!
15) The Answer from the Ashes Magic Trick
This is my all-time favorite mathematical magic trick. Choose a group setting like a big holiday family dinner, an informal teacher conference, a math class, or even a party. You'll need a pencil, two notebook size pieces of paper, a large empty ashtray, some matches or lighter, and a bar of soap.
Preparation: Ten minutes before you begin the trick, excuse yourself and go to the bathroom. There take the bar of soap in hand and with the corner of the moistened end, carefully draw the number 1089 on your forearm. Let it air dry.
When you begin this trick, ask for someone in your audience who believes they have ESP ability and can do addition and subtraction correctly.
Step 1) Tell them to choose any three
digit number where the first and last digits differ by two or
more. Write it down on the first piece of paper.
Step 2) Tell them to reverse the digits and subtract the smaller
number from the larger. For instance, if they initially chose
672 as their number, they would be subtracting 276 from the 672
to get 396.
Step 3) Then tell them to reverse the digits of the difference
they found and now add them together. In our example, you would
add 693 to the 396. (Of course, notice that the total will be
1089...big surprise, huh?)
Step 4) Tell them to double check their work and to show everyone
the final result as you turn your back. Then tell them to write
the final result on the second piece of paper.
Step 5) Tell them to fold up both pieces of paper three or four
times so that you could not see what was written on them.
Step 6) Put both pieces of paper in the large ashtray, ignite,
and burn completely into ashes.
Step 7) Tell your mark to concentrate intensely on the final result
number and tell your audience that you will divine the number
directly from the ashes. Pick up some ashes and rub them directly
onto your soap-written arm. As you continue to rub the ashes,
incredibly the answer 1089 will appear on your arm!!
16) Beat Your Calculator! --- If you are one of those people who feel it is somewhat self-demeaning to use a calculator to do simple two-digit multiplications or other simple mathematical tasks, this page is for you! Go Here!
17) 17 Page --- For those of you who love the number 17, there is an entire web page devoted to it. Here!
18) Numbers Larger Than a Googolplex?--- Well, yes, there are! Quite a few actually...but let me let an expert describe Graham's Number, Knuth's Notation, Ackerman's Function, Conway's Notation, and Moser's Number: Here!
19) The Music in the Numbers --- Did you know that fractal music be produced using pure numbers? Lars Kindermann of the University of Erlangen has produced MusiNum, a program for PCs that does just that. Go Here!
20) The Chronological List of Mathematicians --- There is a listing of every famous, near-famous, and not-so-famous mathematician that ever lived...starting around 1650 B.C. to the present...check it out Here!
21) Newton's Method for Calculating a Square Root --- This method involves a bit of multiplying and dividing but it will arrive at the square over time and trials:
Step 1) Let X be the number you wish to take the square root
of. Let G be your best guess at each stage of the calculation.
Step 2) To find the next G, call it G', you set g' equal to:
G' = ( X + G^2) / 2G
Step 3) This G' becomes your next G. Iterate, that is repeat the calculation over and over, and you will eventually arrive at the square root you seek.
22) What Day of the Week Were You Born On? --- Even though you were there at the moment of your birth, you may not remember exactly what day of the week it was. In fact, not only will it help you find that out, you can find out the day of the week for any date you want in the 1900's. Here is a little trick to help you figure what day that was:
Step 1) Write the last two digits of
the year you were born. Call it A.
Step 2) Divide that number, that is, divide A by four. Drop the
remainder if there is one. Call this answer, without the remainder,
B.
Step 3) Find the month number corresponding to the month you were
born in from the table below. Call it C.
Step 4) Oh, the date you were born on, call it D. (If you were born on the 12 th, call D 12.)
Step 5) Now add A
+ B
+ C
+ D.
Divide this sum by 7. The remainder you get is the key to the
day of the week.
Step 6) In the table of days below, match the remainder with the
day of the week you were born on.
NOTE: This trick will work for any date in the 20th century.
| TABLE OF MONTHS | TABLE OF DAYS | ||
| Sunday = 1 | |||
| January = 1 (0 in leap yr) | July = 0 | Monday = 2 | |
| February = 4 (3 in leap yr) | August = 3 | Tuesday = 3 | |
| March = 4 | September = 6 | Wednesday = 4 | |
| April = 0 | October = 1 | Thursday = 5 | |
| May = 2 | November = 4 | Friday = 6 | |
| June = 5 | December = 6 | Saturday = 0 |
23) Amicable Numbers --- There are a few pair of numbers that have a very peculiar affinity for each other and are so-called "amicable numbers." Take for instance the pair of numbers 220 and 284. It turns out that all the factors of 220, that is those less than itself, add up to 284. And, surprisingly, the factors of 284 add up to 220. I only know of three other pairs like these: 1,184 and 1,210 (discovered by a 16-year-old Italian named Nicolo Paganini), 17,296 and 18,416, and the large pair 9,363,584 and 9,437,056. Can you find others?
24) Incredible Human Calculators --- Oh you might know of someone who can do two- or even three-place multiplication problems in their head, and there are those who can add faster than you can using an electronic calculator, but do you know the stories of Zerah Colburn and Truman Henry Safford?
Zerah Colburn was born in 1804, the son of a Vermont farmer. By the age of eight, he was giving mathematical exhibitions in England where he was asked by a member of the audience to compute 8 to the 16th power. He gave the correct answer 281,474,976,710,656 in about thirty seconds, and brought the astounded audience to tears. Zerah eventually stayed in England, received his formal education there, but strangely his incredible calculating abilities waned as he aged. He died in 1840, at the age of just36, after a life of teaching Greek, Latin, French, Spanish and English in the United States, but not before writing his autobiography in which he outlined his calculating methods.
Another calculating prodigy, Truman Henry Safford, was born in 1836, coincidentally in Vermont. When he was ten, he was given a problem in church by the Reverend H. W. Adam: Multiply in your head 365,365,365,365,365,365 by itself! According to the good reverend's own account, Truman "flew around the room like a top, pullinghis pantaloons over the tops of his boots, biting his hands, rolling his eyes in their sockets, sometimes smiling and talking, and then seeming to be in agony." In less than one minute, though, he had come up with the correct answer: 133,491,850,208,566,925,016,658,299,941,583,225!!! The boy admitted he was exhausted after this calculation. He never did any public exhibitions, went to college, studied astronomy, and like Zerah Colburn, lost much of his amazing abilities as he aged. He died in 1901.
There have been other famous and not-so-famous child prodigies such as John Wallis, the great Johann Carl Friedrich Gauss, André Marie Ampere, George Parker Bidder (Senior and Junior), Johann Martin, Zacharias Dase, Jacques Inaudi, Shakuntale Devi. If you find this area interesting, a little research in the library might prove fruitful.
25) Interesting and Little-Known Algebra and Geometry Facts --- Here are a few helpful and neat little facts that evade most students and teachers of algebra and geometry:
| a) Though everyone can factor a^2 - b^2, a^3 - b^3, a^3 + b^3 and a^4 - b^4, most folks do not know that: |
| a^4 + b^4 = (a^2 + ab(sqrt(2)) + b^2) (a^2 - ab(sqrt(2)) + b^2 |
| b) An approximation exists for the factorial function (for large n) which seems hardly related but works: |
| Stirling's Formula ---> n! ~~ e^(-n) n^n sqrt(2 (pi) n) |
| c) The area of any regular polygon of n sides, each of length x, is given by: |
| Area = (1/4)nx^2 (cot(180°/n)) |
| d) The radii of circumscribed (R) and inscribed (r) circles within such regular polygons are given by: |
| R = (x/2) csc (180°/n) and r = (x/2) cot (180°/n) |
| e) The radius of a circle inscribed within any triangle of sides a, b, and c with semi-perimeter s is given by: |
| r = (sqrt (s (s-a) (s-b) (s-c)) / s |
| e) The radius of a circle circumscribed about any triangle of sides a, b, and c with semi-perimeter s is given by: |
| R = abc / 4 (sqrt (s (s-a) (s-b) (s-c)) |
| f) The perimeter P and area A of polygons (of n sides) inscribed in a circle of radius r is given by: |
| P = 2nr sin(pi/n) and A = (1/2) nr^2 sin (2pi/n) |
| g) The perimeter P and area A of polygons (of n sides) circumscribed about a circle of radius r is given by: |
| P = 2nr tan (pi/n) and A = nr^2 tan (pi/n) |
26) The Magic Tetrahedron (many thanks to R. Leo Gillis) --- Here's a neat trick involving all the numbers from 1 to 26, and three of the five Platonic Solids, the most basic polyhedral shapes. Let's start with the tetrahedron. A tetrahedron, sometimes called a triangular pyramid, is a shape made up of four corners, four triangular sides, and six edges. Since each side is a triangle, it also has a total of 12 angles on its four faces. So the tetrahedron has a total of 26 components, (4 + 4 + 6 + 12 = 26). These components can be numbered from 1 to 26 in a special way.
The basis of the trick is to use three pairs of numbers in
order to create a value for every part of the tetrahedron. The
three pairs are: 1 & 2, 3 & 6, 9 & 18. These six numbers
will be placed on the six edges of the tetrahedron. Each edge
is always directly opposite another edge; that is, if you draw
a line through the center of the object starting from one edge,
you will always reach another edge on the opposite side.
Select any edge and place the number 1 on it. On the opposite
edge place the number 2. Select any of the remaining edges and
place the number 3 on it, and then place the number 6 on the opposite
edge. On the last two edges place the numbers 9 and 18. Now you're
ready to determine all the rest of the numbers, and where they
go on the tetrahedron.
Every corner has three edges that meet at it. Add up the value
of the three edges and give that number to the corner. Every side
has three edges surrounding it. Add up the value of the three
edges and give that number to the side. Every angle on the sides
is formed by two edges meeting there. Add the value of these two
edges and give that number to the angle.
When you are done, you will discover that you have used all
the numbers from 1 to 26 without repeating any numbers! There
are two possible tetrahedra that can be made this way. Can you
find them both?
This trick can also be done on a cube, but without using the
angles of the faces. A cube has six sides, eight corners, and
twelve edges; 6 + 8 + 12 = 26. Using the same three pairs of numbers
we started with above, we can give values to the six sides. Place
the number 1 on a side, and the number 2 on the opposite side.
Place the number 3 on any of the remaining sides, and the number
6 on the opposite side. Then place the numbers 9 and 18 on the
remaining sides.
Just like before, we will add these six numbers together to
create all the others. Every edge of a cube is the meeting place
of two sides. Add the numbers of these two sides and give that
total to the edge. Every corner of a cube is the meeting place
of three sides. Add the numbers of these sides together and give
that total to the corner. Do this for all parts of the cube, and
you will use all the numbers from 1 to 26 without any repetitions.
Unlike the tetrahedron, there is only one way to accomplish this
feat on a cube.
The technique will also work on the octahedron, which has six
corners, eight sides, and twelve edges; 6 + 8 + 12 = 26. In this
case the three pairs of opposite numbers are placed on the six
corners of the octahedron. As before, place 1 and 2 on opposite
corners, 3 and 6 on opposite corners, and 9 and 18 on the remaining
corners. Each of the twelve edges of the octahedron connects two
corners. Add these two corners to find the number for that edge.
Each of the eight triangular sides of the octahedron is surrounded
by three corners. Add those three corners together to get the
value for that side. When completed, you will again have used
all the numbers from 1 to 26 with no repetitions. As with the
cube, there is only one way to accomplish this.
The sharp observer will have noticed that the special set of 3 opposite pairs are used for the edges of a tetrahedron, the sides of a cube, and the corners of an octahedron, which may reveal something important about the relationships of points in space. Also, for the die-hard puzzle solver, the solutions to the above figures may be written in Base 3 numerals, with very interesting results.
27) 500 Digits of e - Named after the world famous mathematician and extreme child prodigy Leonhard Euler, the natural logarithmic base has innumerable applications in all fields of science, business, and mathematics...here is just the first 500 digits or so...
2.71828 18284 59045 23536 02874 71352 66249
77572 47093 69995 95749 66967 62772 40766 30353 54759 45713
82178 52516 64274 27466 39193 20030 59921 81741 35966 29043
57290 03342 95260 59563 07381 32328 62794 34907 63233 82988
07531 95251 01901 15738 34187 93070 21540 89149 93488 41675
09244 76146 06680 82264 80016 84774 11853 74234 54424 37107
53907 77449 92069 55170 27618 38606 26133 13845 83000 75204
49338 26560 29760 67371 13200 70932 87091 27443 74704 72306
96977 20931 01416 92836 81902 55151 08657 46377 21112 52389
78442 50569 53696 77078 54499 69967 94686 44549 05987 93163
68892 30098 79312 77361 78215 42499 92295 76351 48220 82698
95193 66803 31825 28869 39849 64651 05820 93923 98294 88793
32036 25094 43117 30123 81970 68416 14039 70198 37679 32068
32823 76464 80429 53118 02328 78250 98194 55815 30175 67173
28) Interference of Sinusoidal Waveforms Etc. - There is a site that will graphically show you how two sinusoids interact, be they in phase or our of phase, be they the same amplitude or different, or being the same wavelength or different...very instructive...go here. The same teacher that introduced me to this page also has a page to explain "damped" functions, a very important phenomena in virtually every engineering discipline...you can see it here.
29) Fermat's Last Theorem - The Proof - Here is a link to PBS' Nova On Line website that describes the 350-year search for the proof of one of mathematics' greatest mysteries...you can find it here.
30) The Four 4's Puzzle - When I was teaching SAT math, I had occasion to try my hand at this puzzle...the idea is that by using four 4's and any operations, one can write expressions that have the values from 0 to 100 as the answer. Now I finished it and my notes are somewhere in my classroom file cabinet, but I found a place where a solution is listed...so here you go...try your own hand at it before cheating! It's more fun that way!
31) Magic Squares - There are a number of websites dedicated to these little gems...you know the square grids of numbers where the rows, columns, and diagonals all add up to the same number...well, you'll be impressed how far a little concept like that can take you...try here, or maybe here...even here for a ton of magic square links...even find their history is here...
32) The Encyclopedia of Polyhedra - Though I include some very interesting polyhedra texture-mapped with fractals in some of my fractal compositions, and even show a few on seperate gallery pages, I hardly give them the extremely comprehensive attention that George Hart does in his very impressive website...go here...you will find thousands, yes thousands of virtual reality polyhedra to explore and enjoy...lots of very interesting mathematical fun for sure...
33) Chisenbop Arithmetic - Many years ago, I encountered a young Korean student in my class who showed me a set of really neat ways of doing arithmetic using his fingers...multiplying, adding, really quite intriguing, and so simple, well, a child could learn them...it wasn't until maybe ten years later that I heard of it again...it is called Chisenbop...believe or not, there is even a website dedicated to it...here it is...
Chisenbop Multiplying by 9
Hold out your hands in front of you so that your thumbs point
toward one another.
Visualize that your left pinky finger represents 1, the next finger
2, and so on left to right, until your right pinky finger represents
10. Those fingers represent the number you wish to multiply by
9. To do so, simply put the finger down you wish to multiply by
9. All fingers to the left of the down finger represent the tens
digit of the answer while all fingers to the right represent the
ones digit.
Example: 6 x 9. Put the finger representing 6 down (the right
hand thumb). To the left of the down finger, you have 5 fingers
up. That's your tens digit, 5. To the right, you have 4
fingers up. There's your ones digit, 4. Put those together
and you have your answer: 54.
34) Arithmetic Curiosities - Here are just a few interesting patterns in arithmetic that you or your students may explore. Verify these results with paper and pencil or with calculator (if you must):
| 1 x 9 + 2 = 11 | 9 x 9 + 7 = 88 | 9 x 9 = 81 | 6 x 7 = 42 |
| 12 x 9 + 3 = 111 | 98 x 9 + 6 = 888 | 99 x 99 = 9801 | 66 x 67 = 4422 |
| 123 x 9 + 4 = 1111 | 987 x 9 + 5 = 8888 | 999 x 999 = 998001 | 666 x 667 = 444222 |
| 1234 x 9 + 3 = 11111 | 9876 x 9 + 4 = 88888 | 9999 x 9999 = 99980001 | 6666 x 6667 = 44442222 |
35) Earliest Known Uses of Some of the Words of Mathematics - Here is a site that says it all!
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