Question

A large square is divided into adjacent pairs of smaller squares with integer sides, as shown in the diagram (which is not drawn to scale). Each size of smaller square occurs only twice. The shaded square has sides of length 10. What is the area of the large square?

Answer 1

Given : A large square is divided into adjacent pairs of smaller squares with integer sides,  

Each size of smaller square occurs only twice.

The shaded square has sides of length 10.

To Find : area of the large square

Solution:

Notice the bottom half of larger square

There are  8 pairs of squares

Largest is 10  and smallest can be 1

Hence from 1 to  10 only 2 values will be missing

Case 1 :  Let say square above  10 is  8   ( as it can be 8  or 6 only )

Then squares on left side will be  9  

10 + 8 = 18   = 9 + 9    ( height 18 matches )

Adjacent to 8 squares  can be  3 , 3  and 2   ( 3 +  3 + 2  = 8)

8 + 8 +2 + 2 = 10 + 10 = 20  also satisfied

and smallest square would be 1

Square between 9 and 10 = 7

7 + 7 = 14  = 10 + 2 + 1 + 1

Square above 7 must be 4

Lets check bottom edge =  10 + 10 + 7 + 9  = 36

upper edge of half circle

9 + 4 + 4 + 3 + 8 + 8 = 36

Hence all relation satisfies

so  larger square area would be 36²  =  1296

case 2 :  Let say square above  10 is  6

Then squares on left side will be  8  

10 + 6 = 16   = 8 + 8    ( height 16 matches )

Then small square above 10

must be 4 so that   4 + 4 + 6  + 6  = 10 + 10

but then above 4 it would be 1 while that is larger

Hence not possible

so 1st combination is correct

larger square area would be 36²  =  1296

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