[unreality]

I was thinking of doing a diamond shaped Hats-on-Death-Row type puzzle for my "Adventurers of the Braindenizen" topic, and was thinking of how many people would be in the diamond

1 in the first row

2 in the second row

etc

1+2+3+2+1 = 9

I was like "weird, that's 3 squared"

intrigued, I checked it out for 4 people

1+2+3+4+3+2+1 = 16

by this time I was going "!!!" lol

1 = 1²

1+2+1 = 2²

1+2+3+2+1 = 3²

1+2+3+4+3+2+1 = 4²

1+2+3+4+5+4+3+2+1 = 5²

etc

then I thought "well duh, if you flipped the diamond on its side, it would be a 3x3 square"

I dunno, I think it's just interesting how an "addition factorial" like 1+2+3+4 ( =10 ) will always be a square when it scales down again (+3+2+1, total=16), and the square of the apex number no less

I dunno, maybe I'm just a math geek and even more likely, this is probably very well known

anyway, your thoughts?

[Guest]

Well, I do find that intriguing, though squares are the sum of odd numbers. 1+2+3+2+1 = 1+(1+2)+(2+3) = 1+3+5=9

Just putting it out there.

[Brandonb]

I had never thought of it before, but when you think of it, as you said, the diamond is a square on its side. Therefore, the diagonal has the same number of (whatevers) as the length and/or width. Considering this, it makes sense that a square on its side, with however many whatevers across, is going to be that number, squared. Very interesting observation though, especially the way that you came across it. I have definitely never looked at it from that *angle* before.

[Guest]

Oh yeah, that doesmake sense, just adding up the diagonals of a square.

[kingofpain]

If you rotate it and look at it as a square, it incidentally was my first "mathematical discovery".

Bear with me through this (rather boastful) story, but in 4th grade, our evil teacher had required us to memorize all the squares from 1 to 100. This was in India about 13 years ago when teachers had the permission to dish out corporal punishment. He was kind of a jerk who hit you on your open palm with a cane (or slapped the s*** out of you if he forgot the cane) if you didn't do your homework. He was so horrible that we used to hope he had the cane with him so that he wouldn't slap us.

I had heard of Gauss's story (of adding 1 to 100 in a flash) and wanted to be just as cool as he was. So I decided I'll find an easy way to compute squares instead of memorizing like a good boy. I knew the squares from 1 through 10 (through tables), so that was 20 odd numbers right there. I also found out (from a Russian book called "Miracles on Wheels") that to square a number like 65, you simply omitted the 5, did 6*7 and added 25 to the end (4225). So that was 30 numbers I had. For the rest of the numbers, I started racking my head, drawing pictures and stuff. And then the following thoughts poured forth

Take a visual square of symbols (shown with x's and s's) of a random size. If you want to increase its size by one, you replicate the secondary diagonal - which is the length of the sides i.e. square root (this is shown with r's) - and then insert a longer diagonal (shown with d's)

x x s  

x s x	

s x x	


x x s r	

x s r x  

s r x x  


x x s d 

x s d r

s d r x			

d r x x

With this visualization, I struck on the most awesome theorem of my life!: Take a number. Add the number to it's square. Add the next number. Voila! you have the square of the next number! And it worked equally well going backwards! This meant that I was never more that 2 numbers away from the nearest "easy" number (numbers ending in 0 or 5) and a few mathematical operations later, I had square to all numbers from 1 to 100!

I waltzed in confidently to school the next day (we'd had a week to memorize, but I started the night before ) and half the kids were missing. He made us start from 1 and go round with successive squares (even easier for me since I had ample time to prepare for my number) and almost everyone got caned that day but for me and a couple of others. The ones who didn't get caned were picked to compete in a math quiz which I won a small trophy for placing first in

Once I learnt algebra, it became kind of LAME to know that what I did was very boring (basically that n^2 + n + (n + 1) = (n+1)^2), but this was one of those moments in life I will always be proud of!

Cheers!

--

Vig

Edited July 8, 2008 by kingofpain

[Guest]

Awsome work KOP - well done and may yo have many more glorious momets

interesting also is .....

Take any four digit number repeated say (1234) so - 12341234 these eight digit numbers are always divisible by 73 and 137 .. If the eight digit number ends in 0 or 5 then they are divisible by 365 and 50005 too

[unreality]

lol numbers are awesome

[kingofpain]

Awsome work KOP - well done and may yo have many more glorious momets

interesting also is .....

Take any four digit number repeated say (1234) so - 12341234 these eight digit numbers are always divisible by 73 and 137 .. If the eight digit number ends in 0 or 5 then they are divisible by 365 and 50005 too

13 years later, I haven't had too many such moments

Cheers!

--

Vig

[Guest]

I was thinking of doing a diamond shaped Hats-on-Death-Row type puzzle for my "Adventurers of the Braindenizen" topic, and was thinking of how many people would be in the diamond

1 in the first row

2 in the second row

etc

1+2+3+2+1 = 9

I was like "weird, that's 3 squared"

intrigued, I checked it out for 4 people

1+2+3+4+3+2+1 = 16

by this time I was going "!!!" lol

1 = 1²

1+2+1 = 2²

1+2+3+2+1 = 3²

1+2+3+4+3+2+1 = 4²

1+2+3+4+5+4+3+2+1 = 5²

etc

then I thought "well duh, if you flipped the diamond on its side, it would be a 3x3 square"

I dunno, I think it's just interesting how an "addition factorial" like 1+2+3+4 ( =10 ) will always be a square when it scales down again (+3+2+1, total=16), and the square of the apex number no less

I dunno, maybe I'm just a math geek and even more likely, this is probably very well known

anyway, your thoughts?

If you see it logically, you would not have tried all these calculation ... lol

cause your diamond is a titled square

[Guest]

It's still an interesting concept.

[akaslickster]

It's still an interesting concept.

Yes but can you make that 3 dimensional?

[unreality]

I wasn't looking at graphically, just arithmetically ;D ie, the number part of it. Obviously it's a rotated square (I even pointed that out in the OP )

[akaslickster]

I wasn't looking at graphically, just arithmetically ;D ie, the number part of it. Obviously it's a rotated square (I even pointed that out in the OP )

Well, I know a little math but, I can do some outrageous stuff without it. I was more interested in the theory of relativity. I like to learn as I go, instead of being pushed like a dog. I,m a slower slick. My highest scores were in mechanics. Edited July 9, 2008 by akaslickster

[Guest]

not a rule if doesnt work w/ negative numbers ex:

-1+-2+-3+-2+-1 is -9 not 9 which is the square of -3

[Brandonb]

not a rule if doesnt work w/ negative numbers ex:

-1+-2+-3+-2+-1 is -9 not 9 which is the square of -3

Actually that does still hold true. If you take the negative dimensions of a shape, then all that happens is that the sides transpose. Since the sides transpose, then the shape still retains the same positive dimensions. B/c a shape cannot have negative dimensions (unless of course the sides are transposed again). So you're going to end up with the same squared number.

[Guest]

Actually that does still hold true. If you take the negative dimensions of a shape, then all that happens is that the sides transpose. Since the sides transpose, then the shape still retains the same positive dimensions. B/c a shape cannot have negative dimensions (unless of course the sides are transposed again). So you're going to end up with the same squared number.