# Math mini-contest problem for Day 2 on D.Buzz for April 2021 😎

in D.Buzz2 months ago

Math problem for Day 2 😎

Ahmad drew a square (SQ) which has an area of 4 SQ inches. Inside that, he drew a smaller SQ whose corners are at the midpoints of the sides of the larger SQ. If he continues drawing smaller SQs this way, what is the total area of all the SQs?

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# 8 square inches 🎯

## Solution

The first square has an area of 4 square inches. Another square inside it using the midpoints of the first square as the corners will have its sides as √2/2 of the measure of the sides of the original square, which will then give an area which is (√2/2)² or 1/2 of the area of the original square. The drawing of another square continues indefinitely until the area of the last square theoretically reaches zero. Thus, we have an geometric series with an infinite amount of terms.

Using the formula for the sum of an infinite geometric series which is `sum = (first term) * (1 - common ratio)` where we have the first term as 4 sq. in. and the common ratio as 1/2, we get the sum of the areas as 8 sq. in.

# Winner: @jfang003 🏅

The prize of 1 HIVE has been sent to @jfang003's Hive account. 💰

• @eturnerx-dbuzz's answer was correct, but his answer was 40 minutes late from the first correct answer! 😀

Mentions: @holovision, @paultactico2, @dkmathstats, @minus-pi, and @appukuttan66 🤓
Special mentions: @chrisrice, @jancharlest, and @mehmetfix 🤯

My answer is that the total area of all squares if you continue to do so infinitely will be 8 SQ inches.

1st square has each side of length 2. Let this be x.

So the midpoint means that the length of the inner square is sqrt((x/2)^2 + (x/2)^2) = sqrt(2(x/2)^2) = sqrt(2). This means the area of the 2nd square is 2.

If we repeat this pattern, then the area of the following square is 1/2 of the previous square (I did the math and saw that this is true).

Leaving the sum of all squares using this as 4 + 2 + 1 + 1/2 +... and this summation leaves an answer of 8.

• If you mean the total surface area taken by squares the answer: 4 SQ inches. Same as the largest one.

• If you mean adding the area of every new square to be drawn to the previous to. The answer is undeterminable/infinity. A new square can always be drawn.

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Yup! After looking at other entries, I think I missed something: Square areas are converging to zero!

Once again I was wrong, but I love how I came up with my second answer...

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8 sq. inches. I solved intuitively. Each successive square is a diamond inside the outer square, so half the area. Add in the joke about an infinite number of mathematicians walk into a bar the first order a pint, the 2nd half a pint ... the barman pours two pints. Yeah.

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Thanks for the joke, and for trying to answer the Math problem! 😀

As they said, 8. It's 4SUM_n (1/2)^n = 41/(1-1/2)
if... I didn't mess up, what can happens (I'm rusty).

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