Image Credit: VollherzigHerzgeometrie
Mai's friends and lovers were all invited for the Valentine's Day party. After enjoying the wonderful love music and meal, she believed that it's the great idea to play the heart-counting game. The dialogue started:
Mai : The shape is counted as one heart if:
Yugi : That seems quite interesting.
Tea : I am not even good in discrete mathematics, so I am SO bad in counting. Since I believe you are so good in mathematics, you should figure out the best method to answer Mai's problem! There isn't any reason why I would find another man to love! :)
Mai : The couple who arrives with the correct answer first wins the game and earns a grand prize! The time starts now!
[Mai revealed the image as shown above. After few seconds..]
Yugi : I have the answer, and I am very sure it's correct! [Answer hidden]
Mai : We have a winner!
Tea : Wow! What a versed skill you have here! You are the true Valentine mathematician!
How many hearts in all did he find?
Note: Assume that the common circular sectors have same area, and that the 14-sided polygon is regular.
This is the fifth chapter of the story . Check the following chapter directory if you are interested:
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There are 1 4 large sectors and 1 4 small sectors, where the large ones overlap. There are several cases to consider: the heart is formed by either overlapping sectors or tangential sectors.
Tangential Sectors
For small sectors , the approach is trivial. Since there are clearly 1 4 pairs of adjacent sectors that each form a small heart, we count 1 4 possibilities.
For large sectors , since the large circles overlap two segments of a regular 1 4 -gon, known as tetradecagon , then counting the number of hearts formed by two tangential large circles is equivalent to counting the number of sides of 7 -gon or heptagon . In this case, there are 7 possibilities. Thus, since there are 2 regular heptagons enclosed by a regular 1 4 -gon, there are 7 ⋅ 2 = 1 4 possibilities.
So we counted 2 8 hearts of tangential sectors.
Overlapping Sectors
For different-sized sectors , consider the following diagram:
Because there are 1 4 large sectors adjacent to 2 small sectors, we counted 2 8 more hearts.
For two overlapping large sectors , let's now look at the following diagram:
The common region shared by the two sectors indicate the number of hearts, which is congruent to counting the number of midpoints of each segment of 1 4 -gons. Because there are 1 4 midpoints, then there are 1 4 more hearts.
Overall, we counted 4 2 hearts.
Answer
Combining the numbers of hearts from both previous sections, we have 2 8 + 4 2 = 7 0 hearts.