Semi-circles are constructed on AB and BC and drawn in blue, and a semicircle is constructed on AC and drawn in green.

Which curve is longer: blue or the green?

I. Blue
II. Green
III. Equal
IV. Cannot Be Determined

Many American students found this question to be difficult. Can you figure it out?

As usual, watch the video for a solution.

Which Curve Is Longer Green Or Blue?

Or keep reading.

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Answer To Which Curve Is Longer Green Or Blue?

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

A circle’s with diameter d has a circumference πd, so a semi-circle has half that length, or 0.5πd.

The blue curve has a total length of:

0.5π|AB| + 0.5π|BC|
= 0.5π(|AB| + |BC|)
= 0.5π(|AC|)

The green curve’s length is directly calculated as:

0.5π(|AC|)

Thus the two lengths are equal! This is true even if the point B is placed anywhere else along AC, even at the extreme points of A or C.

It is interesting the lengths of the two curves are always in balance no matter where the point B is placed, as the diagram bears a resemblance to the yin and yang symbol that signifies balance.

References

Previous video
https://www.youtube.com/watch?v=JNId5NmvvHg

SAT (undergraduate admissions test)
https://www.businessinsider.com/hard-sat-math-questions-test-practice-2018-3

GRE (graduate admissions test)
https://www.prepscholar.com/gre/blog/gre-geometry-review-practice/

The puzzle is actually problem 37 from the famous 1956 book The Moscow Puzzles.

“A book costs $1 plus half its price. How much does it cost?”

It is a simple puzzle, but it has confused many people on the internet. As usual, watch the video for a solution.

A Classic Puzzle Baffling The Internet

Or keep reading.

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Answer To Price Of A Book Puzzle

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Suppose the book costs b dollars. Then 1 dollar plus half the price is 1 + b/2. Thus we have:

b = 1 + b/2
b/2 = 1
b = 2

The book costs $2.

I did a poll on the MindYourDecisions YouTube channel with the options $1, $1.50, $2, and “the value cannot be determined”. The poll received over 40,000 votes. About 50% agreed with the answer of $2. About 34% said there was not enough information since cost and price can be different. The remaining responses were either $1 or $1.50.

I do think a valid criticism of the phrasing is that cost and price are two different things. Cost is the expense to produce an item while price is the amount a customer pays.

But in this sentence, it should be clear that “cost” and “price” are used synonymously as is often the case in daily life. We say “a movie ticket costs $20” to refer to the price of $20 we have to pay.

Sources

Reddit today
https://www.reddit.com/r/theydidthemath/comments/1vsgrk8/request_i_need_a_second_opinion_on_this_math/
Reddit 5 years ago
https://www.reddit.com/r/theydidthemath/comments/s9k2eb/request_can_someone_do_the_math_i_swear_its_150/

The Moscow Puzzles (problem 37) by Boris A. Kordemsky, edited by Martin Gardner
http://www.math.cmu.edu/~bkell/21110-2010s/algebra-puzzles.pdf
https://store.doverpublications.com/products/9780486270784

On Fyrock’s poll, only 7% gave the answer of $2.
https://www.youtube.com/post/UgkxztB7AMWYjz7rup7eD3-22mVDyYECuVhk

On my community page, about 60% of people give the answer of $2.
MindYourDecisions poll 5 years ago
https://www.youtube.com/post/Ugzv_XryKGRUCCNDluN4AaABCQ
MindYourDecisions poll 2 years ago
https://www.youtube.com/channel/UCHnj59g7jezwTy5GeL8EA_g/community?lb=Ugkx_heZWpPNenkmphzc4IqjSnyU59PCGAeE
MindYourDecisions poll current
https://www.youtube.com/post/UgkxhM0HA5my2PYE7RCDX_1owQIdaoo7oUFZ

My post from 2 years ago
https://mindyourdecisions.com/blog/2024/04/10/price-of-a-book-puzzle-again/
https://youtu.be/0d01sKl0wtY

Singapore model
https://journals.sagepub.com/doi/10.1177/27527263221107526

Cost
https://www.merriam-webster.com/simple/cost

Related memes
https://www.physicsforums.com/threads/math-puzzle.477209/
https://knowyourmeme.com/forums/just-for-fun/topics/14038-you-really-should-be-able-to-solve-this

Square ABCD is divided at its midpoints E, F, G, H. The 4 segments connecting each vertex with the midpoint of the clockwise side bound a region IJKL, which is shaded in the diagram below. What fraction of the square is shaded?

As usual, watch the video for a solution.

What Fraction Of The Square Is Shaded?

Or keep reading.

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Answer To What Fraction Of The Square Is Shaded?

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

We first establish IJKL is a square.

If we rotate the entire diagram by 90 degrees clockwise from its center, then the diagram would remain unchanged. The segments connecting a vertex to the clockwise side’s midpoint are 90 degree rotations of each other, establishing that IJKL has 4 right angles. The sides of IJKL are the respective distances between the segments connecting a vertex to a midpoint, and those distances must be equal so the sides of IJKL are equal. Thus IJKL is a quadrilateral with 4 right angles and 4 equal sides and is a square.

Method 1: similar triangles

Let IL = x.

The right triangle AID is similar to HLD. By similarity, since H is the midpoint on the hypotenuse AD, L must be the midpoint of DI. So DL = IL = x. Triangles DLH and AIE are congruent by symmetry, so AI = x.

We can calculate the area of the square in 2 different ways.

Triangles DIA, AJB, BKC, and CLD are congruent, each with area 0.5(2x)x = x². Square IJKL has an area of x², so the total area of the square ABCD is:

4x² + x²
= 5x²

The fraction of the square shaded is thus:

area(IJKL)/area(ABCD)
= x²/(5x²)
= 1/5

Alternately we could have seen that AID is a right triangle so we have:

AD²
= AI² + DI²
= (2x)² + x²
= 4x² + x²
= 5x²

So once again we have:

area(IJKL)/area(ABCD)
= x²/(5x²)
= 1/5

Method 2: parallelogram area

We will compute the area of the parallelogram DEBG in two different ways.

Let DG = s, so then DH = AH = AE = s. Then we have:

area(DEBG)
= (base DG)(height upon DG)
= (s)(AD)
= (s)(2s)
= 2s²

Now let LK = x. This is a height upon the base DE of the parallelogram. But DE is a hypotenuse of triangle DAE, so we have:

DE²
= DA² + AE²
= (2s)² + s²
= 4s² + s²
= 5s²

Thus we have DE = s√5 and the area of the parallelogram is:

area(DEBG)
= (base DE)(height upon DE)
= (s√5)(LK)
= (s√5)(x)
= xs√5

Setting the two expressions of the area of the parallelogram equal we get:

xs√5 = 2s²
x = 2s/√5

We now find the area of the square IJKL is:

x² = 4s/5

So we have:

area(IJKL)/area(ABCD)
= (4s/5)/((2s)²)
= (4s/5)/(4s)
= 1/5

Method 3: re-arranging areas

There is also an elegant outside the box solution.

We can re-arrange the areas of the small triangles to join with the trapezoids to form a square. (I justify this in detail in the video–but from the above results we would have the new shape has 3 right angles with 3 equal sides, so this new shape would be a square).

The square ABCD has the same area as 5 squares of the size IJKL. Thus it is easy to see the fraction of ABCD that is shaded is 1/5.

References

Reddit AskMath
https://www.reddit.com/r/askmath/comments/1vg9sla/how_to_solve_this/

Variation problem
https://mindyourdecisions.com/blog/2020/05/17/the-square-inside-the-square/

Math StackExchange
https://math.stackexchange.com/questions/2518341/area-of-a-square-inside-a-square-created-by-connecting-point-opposite-midpoint