Jer

Same. I've never seen it in writing before.

Wu-Tang is for the Giants.

Usually, these things from XKCD are either a mixture of real and made-up. In this case, I feel like they're either all real or all made up, but I can't decide which.

This is soooooo embarrassing.

Sometimes it's easier to get directly to the answer 9*(2nd eq.) - 8*(1st eq.) = 10x+9 = 9*19 - 8*13 = 67. Not an easy mental math, so I don't think that's easier in this case.

Sorry typo. sqrt(23) as it is in the Desmos.

The key insight was the line has distances in the ratio 1:2:3 from the three centers.

Desmos | 3D Graphing Calculator

Nice. The line sought is 2x+3y=-6. The two planes are 2x+3y+/-sqrt(26)z=-6 www.desmos.com/3d/k4oxuldkj8

I like the special right triangle windows. I wonder if it would be worth making ones with period/amplitude changes. Also, it looks like you made a cosine grapher as well!

Permutations today: 52! is 8x10^67. The 7th graders lost it.

I remember figuring this rule out on my own back in high school, where to solve a problem, I needed to know which numbers have an odd number of factors.

Thanks for sharing. I was bracing for a challenge, but everything is right there.

So sweet. We had a kitty named Huckleberry who looked a lot this yours many years ago. I'll try and find a photo.

I like math now.

I'd like to say I did that, but I doubled the 27 three times. When I got 216 I realized that would have been easier.

Some "let them eat cake" energy going on there

It's funny because it's true.
(Or the least untrue things he's posted.)

I don't think your argument is sound. There's no guarantee that for any P the triangular cross-section can be adjusted by epsilon and intersect all six sides. On a cube-like shape it does, but for my shape it sometimes can't. Does every F=6, V=8 shape have some P that does work?

Desmos | 3D Graphing Calculator

Looks like you can get a hexagon. At least for this simple case. www.desmos.com/3d/brrufqrvj4

No. A tetrahedron with the tips of two corners sliced off fits the description. I'm having trouble picturing whether it has a hexagonal cross-section.

You guessed correctly. I don't have that one, just the ones that skip by constant amounts.

I think I've cracked it. For any even number of people, there are none (I think). For 7, there are 5. For 9, just 3. (I don't know if I'm missing some sporadic cases, though.)

Another arrangement is acebd, which is its own dual, matching in the same order: ACEBD.

Call your arrangement aedbc, and it matches ADBEC after each rotation pictures. The alternate arrangement adbec matches AEDCB. Swap lower-case letters with upper.

Desmos | Graphing Calculator

www.desmos.com/calculator/r...

It seems like the area is not unique. If the triangle is zero the area is 90.25. If it takes up half the square, the isoceles right triangle has area 361(3-2sqrt2) = 61.94. I think there is a missing constraint based on the Pythagoras tag.

The paper doesn't say every shape checked has the property, it only implies no counterexamples had been found. From the paper:
As of today, most of these polyhedra are known to have Rupert’s property, but some remain open.

My damned backwards brain that still mixes up left and right: ∀ is an upside-down "A" bc it's the indefinite article that means there is one, or it exists. ∃ is a backwards E bc it means "for Every."

Does $200 worth of groceries fill a whole bag, still?

Desmos | Graphing Calculator

Spoiler in the graph. www.desmos.com/calculator/y...