Finally, notice that summing the hex numbers to the cube numbers. To see why this is true, imagine them as three sides of a cube, and then try to imagine putting the smaller ones inside the next largest one, like this: If you can’t see how, try making them with multi-link cubes! Here are some interesting sums of squares: 12 22 + 32 42 = -(1 + 2 + 3 + 4) – – 12 22 + 32 42 + 52 = 1 + 2 + 3 + 4 + 5 – – They are indeed true. To see why, you could use the difference of two squares. Writing the first one as (1+2)(1-2) + (3+4)(3-4) = (1+2)(-1) + (3+4)(-1) and the second one as 1 + (3+2)(3-2) + (5+4)(5-4) = 1 + (3+2) + (5+4) we can see these patterns will continue. Summing the odd squares gives: 12 = 1 = 1 x 2 x 3/6 12 + 32 = 10 = 3 x 4 x 5 / 6 12 + 32 + 52 = 35 = 5 x 6 x 7 / 6 5.3 The great Gauss Here are a few investigations that involve some of the mathematics that German mathematician Carl Friedrich Gauss worked on. 5.3.1 Quadratic Residues* Gauss was the first to study Quadratic Residues in depth. Quadratic Residues are just the square numbers mod n for different n. We found the QRs mod 25 in the second part of the investigation triangle cakes. For example, if you take the square numbers mod 5 you will find that you only get 1 and 4 (as well as 0 which we are not interested in here). We say that 1 and 4 are (QR) mod 5, and 2 and 3 are (NR). quadratic residues non-residues
Finally, notice that summing the hex numbers to the cube numbers. To see why this is true, imagine them as three sides of a cube, and then try to imagine putting the smaller ones inside the next largest one, like this: If you can’t see how, try making them with multi-link cubes! Here are some interesting sums of squares: 12 – 22 + 32 – 42 = -(1 + 2 + 3 + 4) 12 – 22 + 32 – 42 + 52 = 1 + 2 + 3 + 4 + 5 They are indeed true. To see why, you could use the difference of two squares. Writing the first one as (1+2)(1-2) + (3+4)(3-4) = (1+2)(-1) + (3+4)(-1) and the second one as 1 + (3+2)(3-2) + (5+4)(5-4) = 1 + (3+2) + (5+4) we can see these patterns will continue. Summing the odd squares gives: 12 = 1 = 1 x 2 x 3/6 12 + 32 = 10 = 3 x 4 x 5 / 6 12 + 32 + 52 = 35 = 5 x 6 x 7 / 6 5.3 The great Gauss Here are a few investigations that involve some of the mathematics that German mathematician Carl Friedrich Gauss worked on. 5.3.1 Quadratic Residues* Gauss was the first to study Quadratic Residues in depth. Quadratic Residues are just the square numbers mod n for different n. We found the QRs mod 25 in the second part of the investigation triangle cakes. For example, if you take the square numbers mod 5 you will find that you only get 1 and 4 (as well as 0 which we are not interested in here). We say that 1 and 4 are quadratic residues (QR) mod 5, and 2 and 3 are non-residues (NR).
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