Difference of Squares

You have all learned that a2 - b2 = (a + b)(a - b) But perhaps you haven't thought about how to use this to do fast mental calculations! See if you can guess how this trick can help you do the following in your head: 43 x 37 78 x 82 36 x 24 Let's do the first one. 43 x 37 = (40 + 3)(40 - 3) = 402 - 32 = 1600 - 9 = 1591. Practice these, and you'll be able to impress your friends! Presentation Suggestions: See if the class can figure out what you are about to do. The Math Behind the Fact: The moral of the story is if you think CREATIVELY about everything that you learn in mathematics, even the easy stuff like algebra, you will find some surprising applications! See more Fun Facts on lightning arithmetic, or check out the reference below. How to Cite this Page: Su, Francis E., et al. "Difference of Squares." Math Fun Facts. .

Making Magic Squares

Figure 1 Figure 2 A magic square is an NxN matrix in which every row, column, and diagonal add up to the same number. Ever wonder how to construct a magic square? A silly way to make one is to put the same number in every entry of the matrix. So, let's make the problem more interesting--- let's demand that we use the consecutive numbers. I will show you a method that works when N is odd. As an example, consider a 3x3 magic square, as in Figure 1. Start with the middle entry of the top row. Place a 1 there. Now we'll move consecutively through the other squares and place the numbers 2, 3, 4, etc. It's easy: after placing a number, just remember to always move: 1. diagonally up and to the right when you can, 2. down if you cannot. The only thing you must remember is to imagine the matrix has "wrap-around", i.e., if you move off one edge of the magic square, you re-enter on the other side. Thus in Figure 1, from the 1 you move up/right (with wraparound) to the bottom right corner to place a 2. Then you move again (with wraparound) to the middle left to place the 3. Then you cannot move up/right from here, so move down to the bottom left, and place the 4. Continue... It's that simple. Doing so will ensure that every square gets filled! Presentation Suggestions: Do 3x3 and 5x5 examples, and then let students make their own magic squares by using other sets of consecutive numbers. How does the magic number change with choice of starting number? How can you modify a magic square and still leave it magic? The Math Behind the Fact: See if you can figure out (prove) why this procedure works. Get intuition by looking at lots of examples! If you are ready for more, you might enjoy this variant: take a 9x9 square. You already know how to fill this with numbers 1 through 81. But let me show you another way! View the 9x9 as a 3x3 set of 3x3 blocks! Now fill the middle block of the top row with 1 through 9 as if it were its own little 3x3 magic square... then move to the bottom right block according to the rule above and fill it with 10 through 27 like a little magic square, etc. See Figure 2. When finished you'll have a very interesting 9x9 magic square (and it won't be apparent that you used any rule)! How to Cite this Page: Su, Francis E., et al. "Making Magic Squares." Math Fun Facts. .

Some Mathematics facts

If you multiply 111, 111, 111 with 111, 111, 111, you will get the answer 12,345,678,987,654,321.

If you add up the numbers 1 to 1, 000 consecutively, the answer will be 5050.

Among all the 2-dimensional shapes, circle has the largest area but the shortest perimeter.

Prime number is the whole number that can divided by itself and 1 only.　Odd number cannot be divided by 2 exactly whereas even number can be diveded by 2 exactly.

The sum of 2 odd numbers is always an even number. For example, 3+7=10 @ 5+9=14

The sum of an odd number and an even number is always an odd number. for example, 1+2=3 @ 5+8=13

The product of the two odd number and an even number is always an even number. For example, 4×3=12

The difference of the two odd number is always an even number. For example, 5-3=2

Number Divisible by 2

All the even numbers (numbers ending with 0, 2, 4, 6, or 8) are divisible by 2.

For example,

3,698÷2=1849

108,756÷2=54,378

Number Divisible by 3

All the digits in a number.

If the sum of all the digits in a number is divisible by 3, then that number is divisible by 3.

For example,

3,927 ---- (3+9+2+7)÷3=7

therefore, 3927÷3=1,309

180,954 ---- (1+8+0+9+5+4)÷3=9

Therefore, 180,952 ÷3=60318

Francesco Pellos first used decimals in 1942. He used 2.5 to represent two and one over two.

A human hair is normally 0.008cm thick.

The thickness of a sheet of paper is normally 0.012cm.

The Chinese used the idea of negative numbers as early as 200BC and the Indians used it as early as the 7th century. Negative numbers were regarded as impossible numbers in Europe until the 17th century.

Muhammad ibn Musa al-Khawarizmi, a well-known Arabian mathematician in the 9th century, was conferred the title 'Father of Algebra'. He had written a book on algebra entitled 'al-jabr wa'l-Muqabalah'.

Here are a few amazing prime numbers, these prime numbers were proved by the XVIIIth century.

31

331

3331

33331

333331

3333331

33333331

333333331

The next number 333333331 is not a prime number. Whereas it is multipled by 17x19607843.

To multiply by a 2-digit number add the two digits and place the sum between.

For example, 234x11=2574