★ Concept :

Here the concept of Permutations and Combinations have been used. We see that we are given two figures where we have to compare of the order of colouring them is equal. The doubt is raised by the difference in symmetry. But symmetrical difference isn't considered any difference because that can be attained by simply changing the direction. There's even a algebraic proof using Permutations which can give us the number of ways og colouring them. If that's equal, then both figures have same ways of colouring.

Let's do it !

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★ Solution :-

Actually we have 2 ways to prove the required things. Let's discuss each.

Let the figure in the left side be A and let the figure in the right side be B.

• Method I ::

This is a geometric method. This is simply based on rotation. On comparing figure A anr figure B, we see that both are same in structure. Both figures have same colours put in them. And both figures have centered circle with same colour. The only difference between them is that in figure A, purple colour is at bottom whereas in figure B, purple colour is as left. This makes all the colours in fron direction in figure B than in figure A. If we rotate the figure A in clockwise direction to one time so green colour comes at right, then we see that symmetry of both will become same. Thus figures will be same.

This also can be attained by moving figure B in anti - clockwise direction one time so that purple colour comes down. Then too the figures will become equal to each other with same symmetry.

Hence Proved !

• Method II ::

Here we shall use Permutations to prove the given things. We see that both figures have same colours and same number of parts that is equal to 5. So we can calculate the total number of ways to colour the whole figure.

>> Number of ways to colour Figure A = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways

(5 colours and 5 parts and colours aren't repeated)

>> Number of ways to colour Figure B = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways

We see that number of ways to colour both figures is same as 120. This means number of ways to colour the figures is equal to each other thus the colouring pattern is same. Hence both figures are equal with same symmetry.

Hence proved !!

Verified answer

Step-by-step explanation:

So you're in the square above (left image), and want to know how much of your surrounding view is covered by the red square. Using symmetry we know indeed that when you are in the middle, that this is 16

So now you are at the point halfway between the center of the cube and the center of the red square (right upper image). You can use polar coordinates, as your surrounding can be projected onto a concentric sphere embedded around you.

Let's define:

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the polar angle θ=0 in front of you, while behind you θ=π .

the azimuthal angle ϕ=0 above you and ϕ=π2 at your left.

S is the side of the square

A(ϕ)=12Scos(ϕ) is the distance between the center of the red square and the line intersecting the red and yellow plane.

Then the polar angle θ varies over the intersection between the red and yellow square, depending on ϕ :

tan(θ)=A14S=2cos(ϕ)⟹θ=tan−1(2cos(ϕ))

The red area R can be seen as eight times a subarea of the red square, between ϕ=0 to ϕ=14π .

This leads to the following integral:

R=8∫14π0∫tan−1(2cos(ϕ))0sin(θ)dθdϕ

=8∫14π0(−cos(θ)|tan−1(2cos(ϕ))0dϕ

=8∫14π01−cos(tan−1(2cos(ϕ)))dϕ

=8∫14π0(1−14sec2ϕ+1−−−−−−−−−√)dϕ

=8⎛⎝⎜⎜ϕ−cos(2ϕ)+9)√sec(ϕ)tan−1(2√sin(ϕ)cos(2ϕ)+9√)(8sec2ϕ+2)√∣∣∣∣∣14π0

=8((14π−tan−1(13))−0)

=2π−8tan−1(13)

So the fraction compared to the complete surrounding 4π is:

12−2πtan−1(13)

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