Math Puzzle: The sum of all circles

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The chart shows the squares and circles described above with enlarged insets showing smaller and smaller circles in the corner of the blue square.

A red circle is inside a blue square. The layout leaves spaces in all four corners of the square, two of which are filled with smaller circles that barely touch the large red circle and the two corners of the blue square. This, in turn, leaves two smaller spaces in the corners, which are filled with smaller circles, and so on, with smaller and smaller circles ad infinitum. The entire diagram is fit inside a 1 × 1 gray square. What is the total circumference of all the circles?

The total circumference of all circles is π. The circumference of a circle is equal to π times its diameter. So the total circumference of several circles of diameters d₁, d₂, d_3,… East:

πd₁ +πd₂ +πd₃+ … = π(d₁ + d₂ + d₃ +…)

So if we can find the sum of all the diameters, we can multiply that value by π, and we’re done. Due to symmetry, the sizes of the circles do not change if we move some of them to different corners:

The graphic shows the square and circle configuration of the puzzle above, but with a series of smaller circles moved so that they are diagonal to each other and an overlapping dotted line connecting the two corners containing the circles.

Because infinitely many circles tend toward the corners of the blue square, the sum of the diameters of the circles is equal to the length of the diagonal of the blue square (shown as a dotted line). This length is equal to 1 because the outer gray square has a side length of 1.

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abdulmanannet77@gmail.com4 weeks ago

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