Tetracube
The side length of the largest cube that can fit inside a regular tetrahedron whose edges have an unit length of one, can be represented by the formula
A + A B + B B A B ,
where A and B are coprime positive integers. Find A + B .
The answer is 5.
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2 solutions
How did you know that x = 1 + 3 2 + 2 3 s ?
The the problem is by replacing 3 by A and 2 by B from the link below. The solution too by Brilliant Member is from the link below.
link text
[[link text]
If the cube has side x, then the blue triangle has side
x
+
3
2
∗
x
,
and so does the small tetrahedron that it cuts off. Since the difference in heights of the small and large tetrahedra is x, their difference in side lengths is
2
3
∗
x
.
S
o
s
=
x
+
3
2
∗
x
+
2
3
∗
x
.
S
i
n
c
e
s
=
1
,
⟹
x
=
3
+
2
∗
3
+
2
2
2
∗
3
A
=
3
,
B
=
2
,
A
+
B
=
5