4 ms·

Maybe too OT, but here is another proof P = n * (n＋1) * (n＋2) * (n＋3) = n(n＋3) (n＋1)(n＋2) = (nn ＋ 3n) * (nn ＋ 3n ＋ 2) = m (m＋2)

by jonsen 12y ago

Maybe too OT, but here is another proof P = n * (n＋1) * (n＋2) * (n＋3) = n*(n＋3) * (n＋1)*(n＋2) = (n*n ＋ 3n) * (n*n ＋ 3n ＋ 2) = m * (m＋2) If P has an integer square root, it must be (m＋1) Trying (m＋1) * (m＋1) we get m*m + 2m + 1 = m * (m＋2) + 1 = P + 1

arh68 12y ago
I like your proof better! Much simpler to read. The if part confused me a bit, though k = m + 1 P = m * (m + 2) = (k-1) * (k+1) = k^2 - 1
- jonsen 12y ago
  The if part is meant to illustrate the idea, that if P is not a perfect square, then (m+1)^2 will give the nearest perfect square.
ColinWright 12y ago
Or recognize that M x (M+2) is the difference between two squares.
- jonsen 12y ago
  I "see" that m^2 + 2m is almost a square. Just one unit square missing: ---------------- |< m >| | ---------------- | |^| | | | | m*m |m| | | | | |v| ---------------- Is that what you mean?
  - ColinWright 12y ago
    Indeed. As taught, children get shown that a^2-b^2=(a+b)(a-b). This is extremely useful in all sorts of factorizations and calculations. That means that when you see two numbers that are close together you can think of them as (a+b) and (a-b) where b is small, and a is their average. The challenge is to see this in all its variations. We can see that 53^2 - 3^2 = (53-3)(53+3), hence 53^2 = 50x56 + 9 which in turn is 100x28 + 9 which is 2809. So 53^2 is 2809, something I needed a short time ago. It would've taken longer to get a phone, calculator, or computer, than it did to just do it in my head. So (m)(m+2) is the product of two numbers close to each other, hence it's (a+b)(a-b) where a is the average, and b is small. In this case a=m+1 and b=1, so (m)(m+2)=(m+1)^2-1. This is also related to the usual quadratic formula, which is done by completing the square. Same ideas, different disguise.
    - jonsen 12y ago
      The danish minister of finance once impressed the audience at a tv show by multiplying two big numbers in his head very fast. He then explained that this was the method he used.

Maybe too OT, but here is another proof P = n * (n＋1) * (n＋2) * (n＋3) = n*(n＋3) * (n＋1)*(n＋2) = (n*n ＋ 3n) * (n*n ＋ 3n ＋ 2) = m * (m＋2)

by jonsen 12y ago

arh68 12y ago

I like your proof better! Much simpler to read. The if part confused me a bit, though k = m + 1 P = m * (m + 2) = (k-1) * (k+1) = k^2 - 1

jonsen 12y ago
The if part is meant to illustrate the idea, that if P is not a perfect square, then (m+1)^2 will give the nearest perfect square.

ColinWright 12y ago

Or recognize that M x (M+2) is the difference between two squares.

jonsen 12y ago
I "see" that m^2 + 2m is almost a square. Just one unit square missing: ---------------- |< m >| | ---------------- | |^| | | | | m*m |m| | | | | |v| ---------------- Is that what you mean?
- ColinWright 12y ago
  Indeed. As taught, children get shown that a^2-b^2=(a+b)(a-b). This is extremely useful in all sorts of factorizations and calculations. That means that when you see two numbers that are close together you can think of them as (a+b) and (a-b) where b is small, and a is their average. The challenge is to see this in all its variations. We can see that 53^2 - 3^2 = (53-3)(53+3), hence 53^2 = 50x56 + 9 which in turn is 100x28 + 9 which is 2809. So 53^2 is 2809, something I needed a short time ago. It would've taken longer to get a phone, calculator, or computer, than it did to just do it in my head. So (m)(m+2) is the product of two numbers close to each other, hence it's (a+b)(a-b) where a is the average, and b is small. In this case a=m+1 and b=1, so (m)(m+2)=(m+1)^2-1. This is also related to the usual quadratic formula, which is done by completing the square. Same ideas, different disguise.
  - jonsen 12y ago
    The danish minister of finance once impressed the audience at a tv show by multiplying two big numbers in his head very fast. He then explained that this was the method he used.