Tag: problem

Crazy Connections

Graph theory and algebra are required to solve this challenge.

Our challenge begins with a set of points. There are points of red, orange, yellow, green, blue, and purple. These points are all interconnected as follows. Every red point connects to 1 red point, 2 orange points, and 3 green points. Each orange point connects to 1 red point, 2 orange points, 2 yellow points, and 1 blue point. All yellow points connects to 1 orange point, 1 other yellow point, 3 green points, and 1 purple point. Green points all connect to 1 red point, 4 yellow points, and 1 blue point. All of the blue points connect to 2 orange points, 3 green points, and 1 purple point. And finally, purple points connect to 4 yellow points, 1 blue point, and 1 purple point. The question is this: what is the minimum number of points for each colour?

Solution

Solution to Pure Water

We wanted to know how many kilograms of hydrogen Joanne the scientist has. We know that she has 9kg of pure H₂O, no isotopes. The atomic weight of oxygen is 16, and the atomic weight of hydrogen is 1. Since there are 2 hydrogen atoms in H₂O, the ratio of hydrogen to oxygen weight is 2:16, or 1:8. This makes the hydrogen portion of total weight 1/9. Since Joanne has 9kg of water, she has exactly 1kg of hydrogen!

Surprising Non-Congruency

Non-concruent triangles for the challnege.

Solving this challenge requires an understanding of geometry, trigonometry, and angle theorems.

In today’s challenge, we face two triangles who are not congruent. This is surprising because they appear identical, and we are told they share two sides. We can find that they also share and angle. However, because the angle isn’t between the sides, it’s only SSA. SSA is not a valid congruency theorem, because in some cases it gives two possible solutions. In this case, we have both of them, directly connected. The question is this: what is the length of the line that is cut unevenly? That is, the one whose length equals the sum of the sides that the triangles do not share.

Solution

Parallel Pentagon

In today’s challenge, you will need to use trigonometry, angle theorems, and geometry.

Diagram for today’s challenge.

The challenge for today is to find the area of the entire figure. We are given 3 lengths, 2 angles, 1 length equality, an angle equality, and finally, 1 set of parallel lines. Remember the units!

Solution