Checkerboard Area

If both of the large squares have an equal area of 1, then we can simply find the areas of the blue and green areas by figuring out the how much of each square is shaded in.

Since 5/9 of the smaller squares in the first square are shaded in, the area would be 5/9, which is roughly 0.56

Since 13/25 of the smaller squares in the second square are shaded in, the area would be 13/25, which is roughly 0.52

In the end, the blue area is larger than the green area.

The area of white and color is equal except the last color square, the odd one. The blue is bigger than the green, ergo.

What percentage does blue and green occupy?

• blue occupies 5 of possible 9 squares: 55.5%
• green occupies 13 of possible 25 squares: 52%

Blue wins!

Generalizing to any $(2k+1)\times(2k+1)$ checkering with center colored, all the colored areas are larger than $1/2$ . The limit for $k\to\infty$ monotonically goes to $1/2$ . Hence, the coarser the checkering the larger the colored area.

suppose area of large square = 225
so area of blue square = 225 * 5/9 = 125
and area of green square = 225 * 13/25 = 117
Note. 15 * 15 = 225 , 5 is total blue squares, 9 is total squares in LHS square
13 is total green squares, 25 is total squares in RHS square

$\text{as the two squares are identical,their area is same.}$

$\text{so,the blue part has covered}$ $\frac{5}{9}$ $\text{portion of the square}$

$\text{and the green part has covered}$ $\frac{13}{25}$ $\text{portion of the square}$

$\frac{5}{9}=0.56$ and $\frac{13}{25}=0.52$ , so, $0.56>0.52$

so, the blue area is larger.

For a visual solution, first collapse all colored squares to one side:

Then draw a line down the center:

Finally, move as many half pieces from the less filled in side to "empty" spots on the more filled in side. Compare the remaining pieces on the unfilled side.

In both cases, there is one more colored box than white boxes.

The number of colored boxes is $\tfrac 12$ more than half of the total.

The total colored area is therefore half a box greater than half of the total area.

Since the green boxes are smaller than the blue boxes, the green area is less than the blue area.

The ratio of green:white is 13:12. The ratio of blue : white is 5:4 = 15:12 By looking at the ratios you can see that blue>green, 15>13. A no brainier.

Both of the fractions are 0.5 larger than the half of the denominator. Done.

We can assume the two big squares to have side 15 , each of the blue squares would have side 15/3 = 5, hence an area of 25.Whereas each of the green squares would have side 15/5 =3 and area 9.The total blue area would be 25 X 5 , which is greater than the green area, 9 X 13

You don't even need to make the division of 5/9 or 13/25.

Always a fraction divided by a smaller number will be bigger than a same fraction divided by a larger one.

In our case, this applies since 1/9 > 1/25. This since we counted and know that the difference between the squares is 5/9 - 4/9 = 1/9 and 13/25 - 12/25 = 1/25.

Just like 1/10 will be larger than 1/11 every time.

Blue = 5/9 of the total area. Green = 13/25 of the total area. Make the denominator the same to compare fractions, by multiplying 9 and 25. And change the numerator accordingly... Blue, $\frac{5}{9}$ = $\frac{125}{225}$ . Green, $\frac{13}{25}$ = $\frac{117}{225}$ . Therefore the blue is larger.

The colored squares are covering half a square more that half the chess-board. Since the blue squares are larger than the green ones that half extra blue square gives more area than the half green square.

Using basic observations, four squares from green make one square of blue and same goes to the white squares in green making one white square in blue. There are 4 white squares in blue requiring 16 small squares but green contains only 12 white squares and 5 blue squares requiring 20 small squares of green which only has 13, therefore the area of blue is larger.

Blue-White, 15-12 Green-White, 13-12

5/9 > 13/25

If you think about this as a pattern, with infinitely many squares each with more and more tiles than the last and all of them have the tile pattern, you know that as the sequence continues, the area of the colored section approaches 1/2. Knowing this, we can figure out that the green area is closer to 1/2 than the blue. All we have to do is figure out if the area of the blue is more or less than 1/2. And a quick count shows that the area of the blue is 5/9. And because 5/9 is greater than 1/2 and we know that the area of the green square is somewhere between 5/9 and 1/2 we know that the area of the green must also be less than 5/9.

Blue occupies 5/9, green occupies 13/25, multiply numerators by the denominators of the other fraction (to the get numerator when the denominators are the same): 5x25 = 125 for blue, 13x9=117 for green, blue has larger are (common denominator is 225)..

If you take the surface of a green square as a unit then the blue squares occupy 20 units and the green 13 units, the surface of the blue squares is bigger

Blue 5/9 = 125/225

Green 13/25 = 117/225

125 > 117

Blue is 5/9 ,& green is 13/25 on a denominator of 225 , blue is 125 and green is117

Imagine you could shift the colored squares to one side. Do so mentally.

Count the blocks in each of the box's left columns. 2/3 blue, 3/5 green. Or, 10/15 blue and 9/15 green.

I mentally slid the green ones onto each other and noticed there are not enough to reproduce all blue squares. Not the most analytical approach but it worked for this simple example...

Another approach would be to count the green squares and devide the number by four, since one green square equals 1/4 of a blue one. If the resulting number is smaller than five, they can't cover all the blue squares.

here's my take. Both squares are 50/50 color/white after the splitting of smaller and smaller squares approaching infinity. Green is just a few iterations closer to infinity so it's closer to being 50/50. Since blue is "less 50/50", and visually blue has more than white, it must have more area than green does.

Blue = 5/9 = 55% Green = 13/25 = 52%

If a big square area is 1 :

Green area is 13/25

-> wich is half + 0.5/25

Blue area is 5/9

->wich is half + 0.5/9

Since 0.5/9 > 0.5/25 Blue area is bigger !

Blue = 5/9 of the area; green = 13/25 of the area

The squares having equal areas, for example 1.

The area of the blue shaded area from the first square is 5/9 (=0.5 recurring, so 0.555555…).

The area of the green shaded area from the second square is 13/25 (=0.52)

So, the area of blue is larger.

Let's count white squares instead of color: left is 4/9, right is 12/25. 4/9 is equal to 12/27, which is easily comparable with 12/25 and 12/25 is greater. Therefore, white takes more area in the right square and subsequently less is left for the green, and as such, the blue color fills more area than green.

the blue squares represent 5/9th of total area: the green 13/25th. 5/9th is bigger than 13/25th

To find answer without a calculator:

Left square blue to white ratio 5/9 = 4.5/9 + 0.5/9 = 0.5 + 0.5/9 Right square green to white ratio 13/25 = 12.5/25 + 0.5/25 = 0.5 + 0.5/25 Subtract 0.5 from each equation we get left square = 0.5/9 vs right square = 0.5/25 Clearly 0.5/9 (left square) is larger since we're dividing the same number 0.5 by a smaller number ie 9.

It's easier than all of that. The blue square in the left diagram equals 4 unit squares in the right diagram. 5 blue squares times 4 units is 5x4=20 units In the green diagram there are 3 rows of 3 green squares and 2 rows of 2 green squares and so the area is (3x3)+(2×2) = 9+4=13 units 20>13 hence the blue area is greater

Areas equal, given. In blue square, blue larger by 5/9 - 4/9 =1 /9 In green square, green larger by 13/25 - 12/25= 1/25 1/9 is larger than 1/25, Therefore blue has most area.

OK. Look at it from a "mental picture" standpoint. Move all the colored tiles together at the top of each square. You can see that the area of the blue tiles covers 5 tile units on the square. If you do the same with the green tiles and overlay them mentally on the blue tiles, you would be 3 green tiles short to cover all the blue tiles. See! No calculations or calculator!

It will monotonously converge toward 1/2 with increasing lines/rows and the blue is greater than 1/2.

Looking at the white areas made this problem much easier using mental math...

Left white areas: 4/9, Right white areas = 12/25

Left white areas: 12/27, Right white areas = 12/25

With common numerators, Left has a smaller fraction of white areas, so blue has a larger area than green.

Correct answer is: The blue area is larger.

If you assume each green square is one unit x one unit, then the area of a green square is one square unit. There are 13 green squares, so that is 13 square units. The blue squares are then two units on each side, so the area of each blue square is 2x2, or four square units. There are 5 blue squares, so five squares x four square units equals 20 square units. Blue area is larger, 20 square units vs.13 square units.

Divide each large square into 15x15 smaller squares, then calculate how much of these smaller squares are covered by the blue and green areas. For the blue, each blue square is 5x5 = 25, so it's 25x5 = 125 small squares in total. For the green, each green square is 3x3=9, so it's 9x13 = 117 small squares. Hence blue covers greater area.

125:117 blue-to-green ratio

Green: 13/25, Blue: 5/9=15/27. Notice that 15/27=(13+1+1)/(25+1+1). Increasing both the numerator and denominator by 1 grows the numerator faster than the denominator. Thus, the numerator of Green +2 is larger than the denominator of Green +2, and the Blue fraction is greater.

5/9 > 13/25

Let 15 (LCM of 3,5) be the side of larger square.

Blue area = (15/3) x (15/3) x 5 = 125

Green area = (15/5) x (15/5) x 13 = 117

Thus, Blue wins.

Yes, you can arrive at the answ $er by$ calculating relative percentages also.

PS: If x is taken as the side of bigger square:

Area of blue squares = $\frac{5}{9}\times x^{2} \approx 0.556\times x^{2}$

Area of green squares = $\frac{13}{25}\times x^{2} \approx 0.52\times x^{2}$

Thus, Blue wins again! :P

Suppose both squares have unit area. By counting the squares it is easy to see that the areas are

$\text{blue}=\frac{5}{9} \\ \text{green}=\frac{13}{25}$

Now let's use our brains, not our calculators!

$5 \times 25 = 125 \\ 9 \times 13 =117$

and so

$5 \times 25 > 9 \times 13 \\ \implies \frac{5}{9}>\frac{13}{25}$

and so the $\boxed{\text{blue}}$ area is larger.

Another way to see the answer is to recognize the coloured areas as chunky overestimates to $\frac{1}{2}$ . As the subdivision becomes finer the approximations approach $\frac{1}{2}$ from above.