During my search for a different hard-to-find, if not, non-existent set of Pythagorean triples, I found that with x being a natural number, a set of 3 distinct Pythagorean triangles with the same area could be found using the following:

Area: 54x^7 + 189x^6 + 273x^5 + 210x^4 + 91x^3 + 21x^2 + 2x
Using alpha, beta, and gamma; we can create the m and n to be used in Euclid's formula that will give us our triangles.
a = m^2 - n^2 , b = 2mn, c = m^2 + n^2

a^2 + b^2 = c^2 and Area = (a•b)/2

alpha(x) = 3x^2 + 3x + 1
beta(x) = 2x + 1
gamma(x) = 3x^2 + 2x
For T1, m1 = alpha(x) and n1 = beta(x)
For T2, m2 = alpha(x) and n2 = gamma(x)
For T2, m3 = beta(x) + gamma(x) and n3 = alpha(x)

T1:
m1 = alpha(x)
m1 = 3x^2 + 3x + 1
n1 = beta(x)
n1 = 2x + 1
a1 = (m1)^2 - (n1)^2
a1 = (3x^2 + 3x + 1)^2 - (2x +1)^2
a1 = (9x^4 + 18x^3 + 15x^2 + 6x + 1) - (4x^2 + 4x + 1)
a1 = 9x^4 + 18x^3 + 11x^2 + 2x
b1 = 2•(m1)•(n1)
b1 = 2•(3x^2 + 3x + 1)•(2x +1)
b1 = (6x^2 + 6x + 2)•(2x + 1)
b1 = 12x^3 + 18x^2 + 10x + 2
c1 = (m1)^2 + (n1)^2
c1 = (3x^2 + 3x + 1)^2 + (2x +1)^2
c1 = (9x^4 + 18x^3 + 15x^2 + 6x + 1) + (4x^2 + 4x + 1)
c1 = 9x^4 + 18x^3 + 19x^2 + 10x + 2

T2:
m2 = alpha(x)
m2 = 3x^2 + 3x + 1
n2 = gamma(x)
n2 = 3x^2 + 2x
a2 = (m2)^2 - (n2)^2
a2 = (3x^2 + 3x + 1)^2 - (3x^2 + 2x)^2
a2 = (9x^4 + 18x^3 + 15x^2 + 6x + 1) - (9x^4 + 12x^3 + 4x^2)
a2 = 6x^3 + 11x^2 + 6x + 1 = (x+1)(2x+1)(3x+1)
b2 = 2•(m2)•(n2)
b2 = 2•(3x^2 + 3x + 1)•(3x^2 + 2x)
b2 = (6x^2 + 6x + 2)•(3x^2 + 2x)
b2 = 18x^4 + 30x^3 + 18x^2 + 4x
c2 = (m2)^2 + (n2)^2
c2 = (3x^2 + 3x + 1)^2 + (3x^2 + 2x)^2
c2 = (9x^4 + 18x^3 + 15x^2 + 6x + 1) + (9x^4 + 12x^3 + 4x^2)
c2 = 18x^4 + 30x^3 + 19x^2 + 6x + 1

T3:
m3 = beta(x) + gamma(x)
m3 = (2x + 1) + (3x^2 + 2x)
m3 = 3x^2 + 4x + 1 = (x+1)(3x+1)
n3 = alpha(x)
n3 = 3x^2 + 3x + 1
a3 = (m3)^2 - (n3)^2
a3 = (3x^2 + 4x + 1)^2 - (3x^2 + 3x + 1)^2
a3 = (9x^4 + 24x^3 + 22x^2 + 8x + 1) - (9x^4 + 18x^3 + 15x^2 + 6x + 1)
a3 = 6x^3 + 7x^2 + 2x
b3 = 2•(m3)•(n3)
b3 = 2•(3x^2 + 4x + 1)•(3x^2 + 3x + 1)
b3 = (6x^2 + 8x + 2)•(3x^2 + 3x + 1)
b3 = 18x^4 + 42x^3 + 36x^2 + 14x + 2
c3 = (m3)^2 + (n3)^2
c3 = (3x^2 + 4x + 1)^2 + (3x^2 + 3x + 1)^2
c3 = (9x^4 + 24x^3 + 22x^2 + 8x + 1) + (9x^4 + 18x^3 + 15x^2 + 6x + 1)
c3 = 18x^4 + 42x^3 + 37x^2 + 14x + 2

​

https://preview.redd.it/ccuquhv7zrrc1.png?width=477&format=png&auto=webp&s=cb9a6e77cf06cc2766826f6bac1139aa505631de

This seemingly simple question on a retail management (grocery store) exam took me longer than I expected. My notes were half a page long. I assume I'm just bad at math and missing a shortcut to solving it faster.

However, I also think it's phrased in such a way to make it confusing on purpose?

Question:

A box of cherries is $6.47. The current in-store sale promo: Buy Three, Get One Free (total of 3 boxes for the price of 2).

A customer has a coupon for 35% off any purchase at the store. The coupon can only be applied to the original price of the item. It cannot be applied as a discount to any in-store sale prices or promotions.

The customer buys four boxes. They have two options for payment:

1. They can use the Buy Three, Get One Free promotion on the first three boxes, and apply the 35% coupon toward the cost of a fourth additional box.

2. They can use the 35% off coupon for all 4 boxes at their original price, but cannot receive the Buy Three Get One sale.

Which of these options saves the customer more per box of cherries?

I’m doing an egg hunt for my astrophysicist brother and would like to make him work for it with an equation that, when solved, either spells “Piano” or “egg”. Can anyone help me with that?

I understand how to find the area of squares and rectangles. A = L x W = X² BUT, I can't even BEGIN to understand this problem. There's a plot, there's a lawn, there's a garden. It's so convoluted that I truly can't even begin to try and solve this. Please help me. Here's a link to a photo of the problem through imgur: https://imgur.com/gallery/rdC9A3M

The above question was posed to the asksciencediscussion sub by some physics student this morning and (correctly) deleted after a few hours. Before it died, I managed to find an elegant solution, and I was pretty proud of myself. It's basic statics, but if you haven't done physics in a while it's a fun and approachable refresher. I think it has enough merit to deserve a casual look.

Hints: >!One of the responders thought it was necessary to set up a system of equations to prove the result, but it's not. Also, the 700N is just a MacGuffin that sets it up as an inequality. Finally, Fa+Fb+Fw=0; if it didn't, the intersection point would move.!<