Combinatorial Geometry!!

This is an interesting problem I saw in an NMTC paper.... A solid cube is chopped off at each of it's 8 corners to create an equilateral triangle with 3 new corners. All possible diagonals are drawn. Find the number of these diagonals which completely lie inside the new 3-D body.

#Geometry #Combinatorics #MathProblem #Math

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

The truncated cube has $3\times8=24$ vertices. Each vertex $v$ belongs to two of the octagonal faces and one triangular face. Thus there are $7+7-1=13$ vertices to which $v$ can be joined which result in a nontrivial diagonal which lies on the surface of the truncated cube. Thus there are $10$ vertices to which $v$ can be joined forming a nontrivial diagonal lying inside the truncated cube.

Thus there are $\tfrac12 \times 24 \times 10 = 120$ nontrivial diagonals lying inside the interior of the truncated cube.

Mark Hennings - 7 years, 7 months ago

Note that no diagonal lies outside the truncated cube. Therefore the total number of diagonals inside the truncated cube is $\binom{24}{2}=300$ . But we counted edges and the diagonal of each octagonal face with that too. We subtract the $36$ edges and the $120$ diagonals of all the octagonal faces to get $300-36-120=\boxed{144}$ . For some reason I got a different answer than Mark, so who made the mistake?

Daniel Liu - 7 years, 7 months ago

${24 \choose 2}=276$

Mark Hennings - 7 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$