Unit Cubes 2 – 5 X 5 Corners

Probability Level 2

If someone painted the outside of a $5 \times 5 \times 5$ cube made out of $1 \times 1 \times 1$ unit cubes, how many of the unit cubes would have paint on exactly 3 sides?

The answer is 8.

9 solutions

Sharky Kesa
May 23, 2014

For any $n \times n \times n$ cube, where $n$ is greater than 1, there is always 8 cubes with 3 faces painted, namely the corners.

This problem was so easy that I suspected it was some trick question

Ceazar Sunga - 7 years ago

Yeah, I was sitting there trying to think of another answer. lol

Revanth Gumpu - 7 years ago

Yes and I still don't get why 60% people got this wrong

Aryan Gaikwad - 6 years, 3 months ago

It is always 8 cubes in such a problem with three sides painted.

If X be the length of initial large cube and x be the length of small number of cubes then,

No. of cubes with no paint = {(X/x) - 2)^3 No. of cubes with 1 face painted = 6 * {(X/x) - 2}^2 No. of cubes with 2 adjacent faces painted = 12 * {(X/x) - 2)

Nishant Prabhu - 7 years ago

Can you prove that, the no.of cubes with no paint={(X/x)-2}^3?

Chitranjan Singh - 1 year, 1 month ago

yeah, just a trick!

Nguyễn Thị Hồng Miên - 7 years ago

Ok. Now how many are unpainted?

BHinstitute Harendran B - 7 years ago

Number of cubes with any paint is 27. The formula is {(5/1) - 2}^3

Nishant Prabhu - 7 years ago

27 are unpainted

GOVIND PHATAK - 7 years ago

if by unpainted you mean not painted on all three sides but painted none the less that would be total - 8 (obviously) meaning 15617. Now how about the number that are not painted at all? That would be 15475

Muhammad Khwaja - 7 years ago

I misunderstood this problem like this, when n n n cube has been painted on the 3 faces, how many unit cubes was painted? so my solution is 61..

Myung Chul Lee - 7 years ago

But there will be only 750 faces totally how can 15k odd faces be unpainted

Sankarshan Bhat N - 6 years, 7 months ago

hid one was easy !

Andreas Pauper - 3 years, 1 month ago

A Former Brilliant Member
Jun 17, 2017

The only cubes that are painted on three sides are the corner cubes. There are a to total of $8$ corner cubes. So the answer is $\boxed{8}$ .

Pranitha Deverachetty
May 25, 2014

It's easy peasy lemon sqeezy! We just have to count all the corner pieces and write the answer.

It's not csgo.....

ameez khan - 4 years, 8 months ago

Kawaii Kuma
May 24, 2014

Look at the corner =.=". That's all!

only the corner cubes are exposed from all three sides thus there are only 8 coner cubes which can be painted from three siudes

Akhtar Javed - 7 years ago

That's what my comment is about :-?

Kawaii Kuma - 7 years ago

Zaheer Ahamad
Oct 26, 2014

Its just 8 . since we get only corner piece with 3 face paint

Ajinkya Shivashankar
Nov 14, 2016

Well,the number of cubes having paint on exactly 3 "faces" is certainly 8, However, number of cubes having paint on exactly 3 "sides" is 0.

Mxjd Ultimate
Jun 7, 2021

The pnly place where are 3 sides showing are the corners which are 8 of them

Bostang Palaguna
Aug 16, 2020

they're the corners

Jessie Trowbridge
Aug 12, 2015

Well.... if you look at the picture ... the only one with 3 sides is the cornors. So count the corners..

Unit Cubes 2 – 5 X 5 Corners

The answer is 8.

9 solutions

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