In this case, the algebraic proof is also simple enough: For any integer n, we have 2n+1= 2n+1 + n^2 -n^2 = (n+1)^2-n^2.pic.twitter.com/tQ35a5iFXT
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In this case, the algebraic proof is also simple enough: For any integer n, we have 2n+1= 2n+1 + n^2 -n^2 = (n+1)^2-n^2.pic.twitter.com/tQ35a5iFXT
Indeed. But it isn't half so pretty.
But which odd integer is the smallest one that is the sum of two cubes in two different ways?
1729 10³+9³ 12³+1³ There's a famous story about GH Hardy going to see Ramanujan in a cab numbered 1729, and he said he wished it'd been more interesting a number, and Ramanujan said, "nonsense. It's the smallest imteger that can be expressed as the sum of two cubes in two ways"
Also the fact that sum of consecutive odd numbers = squares of numbers ex. 1 + 3 + 5 = 9 can be proved from the last grid.
Can you expand on this one? Don't quite understand it
Or equivalently sum of n odd numbers from 1 to 2n-1 is a square of n. n^2 = 1 + 3 + ... + (2n-1) .To get the above clamin you just substract two formulas with n+1 and n. So (n+1)^2-n^2 = 2n+1
Hydraulics!
I need more help
Those seem to be single tiles made up of two parts, so the odd number here is 5. Odd numbers are just evens (a number divisible by 2) + 1: 5=2*2+1. The 1 makes a corner position. So 5 is like (2+1)*(2+1)-2*2 (the 2*2 are the 4 tiles that fill the rest of the square).
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