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August 23rd, 2012, 05:43 PM   #1
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Two Multiplication Table Fun Facts!

One I learned about in a neat little popular math book called The Book of Numbers by Conway and Guy, The second something I noticed myself. I claim originality, but sincerely doubt priority, especially since it is really pretty trivial!

Okay, now I know from past bitter experience that I can't format rows and columns to save my life. But picture standard multiplication table, but just the products, as in:


1 2 3 4 5 6
2 4 6 8 10 12
3 6 9 12 15 18
4 8 12 16 20 24
5 10 15 20 25 30
6 12 18 24 30 36

Now, the upper leftmost square in the grid is just 1, and 1 =1 (with me so far? )
The next larger, ie 2X2, square is made by adding the unit squares with values 2, 4 and 2, and 2+4+2=8
The next larger square is made by adding the unit squares 3, 6, 9, 6 and 3 and 3+6+9+6+3=27

Ok, so what is the pattern? Will it continue to hold, and why?

Take any number anywhere on the multiplication table and add it to all the numbers above and/or to the left of it. That sum will always be the PRODUCT of two ____ numbers. Fill in the blank and explain why!
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August 23rd, 2012, 05:48 PM   #2
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Re: Two Multiplication Table Fun Facts!

To make the second question clearer. If we chose the 12 = 3*4, we would add 12 to the numbers to its left, ie 9, 6, 3 and the numbers above it,ie 8, 4 AND all the numbers that are both above it AND to its left, ie 1, 2, 3, 2,4, 6, all of which adds up to 60.
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August 23rd, 2012, 06:03 PM   #3
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Re: Two Multiplication Table Fun Facts!

You can actually explain the first fact in terms of a set of special cases of the second fact.
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August 24th, 2012, 04:46 PM   #4
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Re: Two Multiplication Table Fun Facts!

1, 8, 27. 64, 125, ... stop me when you recognize this sequence! ... 216, 343, 512 ...
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August 25th, 2012, 02:24 AM   #5
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Re: Two Multiplication Table Fun Facts!

Quote:
Originally Posted by johnr
1, 8, 27. 64, 125, ... stop me when you recognize this sequence! ... 216, 343, 512 ...
1 = 1^3

8 = 2^3

27 = 3^3

64 = 4^3

125 = 5^3
.
.
.
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August 25th, 2012, 03:50 AM   #6
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Re: Two Multiplication Table Fun Facts!

Yes, Balarka! So relate that back to the multiplication table!
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August 25th, 2012, 04:26 PM   #7
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Re: Two Multiplication Table Fun Facts!

Quote:
Originally Posted by johnr
Yes, Balarka! So relate that back to the multiplication table!
Ok, how about THIS, he continues with all the mock enthusiasm he can muster ...

Triangular numbers can be expressed as the sum of the first n consecutive integers. So 10 = 1+2+3+4 and 6 = 1+2+3

So, if you express two triangular numbers in this sort of form and then multiply them, what do you get? A rectangular slice of the regular multiplication table, right? So what does that slice's numbers add up to? the product of two triangular numbers, no? Yes? YES!!!

Ok, so triangular numbers can ALSO be expressed as ((k)k+1))/2 Two CONSECUTIVE triangular numbers can be expressed as ((x+1)(x+2))/2 and ((x)(x+1))/2

If you square both and find the difference, voila, it equals (x+1)^3 Yes, it'l always be a cube.

But why square them and take the difference? Because, each square section of multiplication table sums to the product of two triangular numbers, as just noted, but of course in this case it will be the SAME triangular number multiplied by itself, hence a square. SO the DIFFERENCE between the sum of a larger square and the immediately smaller square on the multiplication table will always equal a cube related to the two squared triangular numbers in a specific way. So 10 = (4*5)/2 and 6 = (3*4)/2 and 10^2 = 6^2 = 64 = 4^3

But the difference between these two square patches will ALSO equal the sum of the numbers in the larger square and NOT in the smaller square. Hence, you will find the cubes on the successive outer"ledges" of the squares on the multiplication table:

1 = 1= 1^3
2+4+2 = 8 = 2^3
3+6+9+6+3 = 27 = 3^3
4+8+12+16+12+8+4 = 64 = 4^3

und so weiter ...

Of course, this is not the only possible proof ...
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