If you had a pile of sticks (lines) of random length could you always make a triangle out of three of them? If not, can you figure out a rule for whether they can or not? What about making a four-sided figure from four sticks? Do the same rules apply? Scroll down for a hint. Solution at the bottom. - - - - - - - - - - - - - - - - - - - - - - - - Hint: No, you can't take any three random sticks and make a triangle or four-sided figure out of them. There is a specific relationship between their lengths that must be true. Scroll down for the answer. - - - - - - - - - - - - - - - - - - - - - - - - - - - - Answer: For a triangle, the combined length of the two shorter sticks must be longer than the length of the longest one. To put it on more mathematical language is 0 < a < b < c then a + b > c to make a triangle. This is called the Triangle Inequality. For a four-sided figure the answer is similar, the sum of the three shortest sides must be bigger than the longest side.

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You are building something awesome with Lego and need two bricks that are the same color. Because you are so captivated by your creation, you reach to grab them from a pile that has three different colors of the size brick you need without looking at them. How many bricks do you need to grab to ensure that you have two of the same color? What if you need three of the same color? Advanced: Can you figure out a general formula that tells you how many you need to grab given the number of same color bricks you need? What if we changed the number of colors in the pile? Let's say that there are four colors in the pile and you need two of the same color, how many would you need to grab to get two of the same color in that case? Mega Advanced: What is the probability in the first situation that you could grab two of the bricks from the pile without looking and get the the same color? Good luck and I'll post the rest of the answers soon! The first two answers are below - - - - - - - - - - - - - - - - - - - - - - - - - - Answer: You might be tempted to think that you could just try your like and just take two, but we want to be absolutely certain. Let's think about the worst case that could possibly happen - you grab three and they are all the same color. If you take one more you will be certain to have two matching bricks, so you need to grab four. If you need three of the same color, you can think the same way. It's possible to grab six bricks and have two of each color if you are unlucky, but if you grab one more, that is seven, you will be sure to have three of the same color. This is an application of something mathematicians call The Pigeon Hole Principle. It says that if you have N pigeon holes and N+1 pigeons, you can be certain that there is at least one hole that has more than one pigeon in it. Sounds kind of silly and obvious but it turns out it to be incredibly useful in math.

One of my favorite things is when art and science intersect to create something new and unique. Scientists recently discovered that the three thousand year old art of paper cutting called Kirigami can be used to create a better parachute. I've included a template that you can print out on a piece of construction paper to create your own. The template is just one way to make one and is provided to familiarize you with the basic parts. Here's a link to the original video of the scientists that came up with this https://youtu.be/6rrDW6YIbXI

Draw the wildest pattern of shapes you can. No matter how crazy your drawing, you can always color it in using four colors without any of them touching each other. Don't believe me? Try it out for yourself. This idea was first discovered 1852 and to this day no human has proven that it's true nor found a counter-example showing that it's false. Though in the 1960's a computer was used to prove it is true, it used methods that some people don't fully accept because a human can't go back and check it. Either way, this was historically the first conjecture that was tackled by a computer.