60 of 100: Letters Assemble!

A - Working Solution

B - Only possible approach leaves it unsolvable

C - Two possible approaches both leave the grid unsolvable

Rather than just try to find a solution, we can look at the different shapes and see if we can rule any of them out right away.

The easiest one is B. Looking at the one block that is sticking out at the top, we see that there isn't anything that can fill it in, so B can't be assembled using these pentominoes.

Looking at C, it shouldn't take long to convince yourself that there isn't a place to put the L-shaped pentomino that will leave room for the other two pentominoes.

The only one we haven't ruled out is A, and, through some trial and error, a solution can be found for it!

It is easy to notice that it is impossible to assembly figure B or figure C.

First thing that comes to mind is although not exclude some two options options but just assembly some figure by yourself, in your imagination , that also can be done due to small complexity of those figures.

That is a typical IQ test task.

So we either just solve it or we are not "gifted" enough.

I wonder if we can find a way of describing the shapes algebraically. One method seems to be to describe each of the basic shapes possibly by a 3 x 3 matrix of ones and zeros, find some invariant property of each of these matrices, for example, the characteristic polynomial or the eigenvalues, find the characteristic polynomial of each of the candidate solutions & see if they could be derived from the polynomials of the basic motifs.

This is a tentative exploration of how algebraic and analytical methods could be used to solve such supposedly purely geometric problems. This problem may be solved through simple visualization but analytical / algebraic methods would be helpful when the problem is scaled up in complexity and / or when it is easy to make a mistake

So A is the answer

Shape U requires no rotation or reflection.

Shape Z must be rotated 90 degrees clockwise, though no reflection is needed.

Shape V is rotated 45 degrees anti-clockwise, though no reflection is needed.

The only letter that the shapes can fit in is a.

Plate C is possible but you have to use U twice and without using V.

Plate B is not possible since there's one stud at the top.

Put U the way it is configured so that left two squares fit into left squares of A. Then fit Z on top of U by rotating it to the right 90 degrees counterclockwise. Finally rotate V counterclockwise 45 degrees and put it in the bottom and right squares of A which are below U and to the left U and Z.