Jan 25, 2021

Providing Madison children with a friendly environment that fosters learning beautiful mathematical theories beyond the regular school curriculum and developing critical thinking and problem solving skills.

# Pick's theorem

## by Connor Simpson

Pick's theorem relates the area of a polygon whose vertices lie on points of an evenly spaced grid to the number of grid points inside it. We'll do a sequence of examples to discover this theorem, outline a proof, and consider 3-dimensional analogues.

# Math Circle is Back!

## (virtually for now)

After a semester off, we are excited to wake from our slumber! Until we can go back to normal in-person meetings, the Madison Math Circle will operate in a reduced format during the semester.

We will have a monthly Zoom talk on the first Monday of every month (February through May). It's a lot less than before, but do not fret! On weeks without a talk, we will send out a lovely math-filled newsletter — just like this one! Each newsletter will have a riddle or two for you to grapple with and a fun math video to watch. Send in your solutions to the riddles to reveal our solution and for a shout-out in the next newsletter.

Looking forward to a great semester of learning!

– Professor Andrews

# Video of the Week

Remark: The last part of the video contains some material requiring some knowledge of trigonometry. The first 20 minutes needs no background.

# Weekly Stumper

## (This is a very open-ended problem. Find the best procedure you can, even if you don't know it's the best possible.)

Hancock tower in Chicago has 100 floors. You want to find the highest floor of Hancock tower from which you can drop an ostrich egg without it cracking.

For each of the following cases, find the best procedure that you can to finds the right answer. We are measuring "best" by trying to minimize the number of egg-drops that the procedure might take in the worst-case scenario. So, we consider a procedure which might give the right answer in 1 drop, but might also take 100 drops to be less good than a procedure which always takes 90 drops.

1. You have exactly one ostrich egg
2. You own an ostrich farm (so, you have an unlimited supply of eggs)
3. You have exactly two ostrich eggs

Example for a “bad” procedure for case 1 that may not result in the exact correct answer: drop your single egg from floor 100. If it doesn’t break, then floor 100 is the highest floor from which you can drop an egg without cracking it, but if the egg breaks, then you are out of eggs and did not find the answer.