These sites got big a couple years ago, and there are some great articles at codinghorror.com about how they are a scam and basically a lottery.

This would be fine if the sites were regulated like a lottery. But in fact, they advertise on TV as legitimate auction sites.

Here’s a video that summarizes some of the issues with penny auctions:

Video: Don’t Waste Your Money – Penny auction scams

Some of the main objections to these sites are:

• the timers are different: the auction time keeps increasing as people bid
• it costs money to bid, and often you end up with nothing
• there may be costs just to sign up

Another issue to add is:

• these sites often offer free bids for people to sign up. These free bids increase the length of the auction (as the timer resets) which forces paying bidders to have to place more bids, generating extra revenue for the site

The bottom line: avoid penny auctions unless you are looking to do something similar to gambling and playing the lottery.

But what happens when you cut the pizza in odd ways, like the following division?

Let’s play a game to test your skill at getting the most pizza.

A pizza dividing game

You and I are to share a round pizza, perfectly shaped as a circle.

I will slice the pizza by making cuts from the middle to the crust. The slices may be of various sizes.

We agree to divide up the pizza according to the following rules of politeness:

a) We will pick pieces in alternate turns

b) You get to pick the first slice

c) Only slices that are adjacent to already picked slices can be chosen. That means after the first turn, there will be two available slices to pick on each turn

If I make the pizza an even number of slices, then we both end up with the same number of slices.

If I make the pizza an odd number of slices, you end up picking the first and last slice, so you end up with one more piece of pizza.

If we are both playing strategically, would you rather have me slice the pizza into an even or an odd number of slices?

Think about this carefully–finding the answer is a lot harder than it sounds! Scroll down below to read the answer.
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The solution to the game

The intuitive answer is that you are better off with an odd number of slices as you end up with one more piece of pizza. But it turns out that having more pieces is not to your benefit.

You are actually better off picking an even number of slices!

With an even number of slices, you can always guarantee that you get at least 1/2 of the pizza.

Why is that?

The proof is simple. Imagine the pizza slices are colored as red and green in alternate slices. There will be an equal number of red and green slices. By the pigeonhole principle, either the red or green slices will make up at least 1/2 of the pizza. You as the player going first can guarantee the larger set by picking the first slice of the right color. I will be forced to pick an adjacent slice of the opposite color, and in subsequent turns you can always pick the slice I just revealed of the right color, forcing me again to pick a slice of the opposite color. You’ll end up with all the slices of the right color and get at least 1/2 of the pizza.

(Also check out a similar problem: the coins in a row puzzle)

With an odd number of slices, I can actually cut the pizza so you only get 4/9 of its area.

How is that possible?

The math required to prove this is quite complicated, and I will refer you to two papers that provide detailed proofs. See How to eat 4/9 of a pizza and Solution of Peter Winkler’s Pizza Problem.

Here’s an example of a pizza in which the first person picking the slice is limited to 4/9 of the pizza (image from this paper).

The numbers represent the size of the slices and add up to 9. There are slices of size “0,” but we can also interpret that to mean really, really small slices.

Suppose Alice picks first, and Bob picks second. Here is the strategy Bob can use to limit Alice’s pizza total to 4/9:

If Alice starts with a non-zero piece, Bob picks the available piece adjacent to a thick cut. Afterwards Bob always picks the piece just revealed by Alice (so he follows Alice) unless this would mean eating from a still untouched interval. If both pieces available to Bob are from untouched intervals, he picks the piece from the interval of smaller size. One can verify (several elementary cases) that Bob always eats at least 5 with this strategy.

It’s counter-intuitive, but you’re best with an even number of slices: you end up with the same number of pieces, but you can control that you end up with more pizza.

Here is the puzzle:

This problem is about two really, really long ropes A and B.

Rope A is long enough that it could wrap around the Earth’s equator and fit snugly, like a belt (let’s say 25,000 miles).

Rope B is just a bit longer than rope A. Rope B could wrap around the Earth equator from 1 foot off the ground.

How much longer is rope B than A?

(assume the earth is a perfect sphere)

Extension: let’s say that rope C can wrap around an equatorial line for a sphere that’s as big as the planet Jupiter (about 273,000 miles). Rope D is just a bit longer, and it can do the same thing from 1 foot off the ground.

How much longer is rope D than C?

Can you solve it? Give it a try before reading the solution below.
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The answer to the puzzle

The surprising part is that both questions have the same answer!

To see why, suppose that r is the radius of the Earth. Then the enlarged circle has a radius of r + 1.

We can calculate how much longer rope B is by subtracting the difference of the circumferences of the two circles:

2 π (r+1) – 2 π r = 2 π = about 6.28 feet

Therefore, rope B is longer by 6.28 feet.

But notice the remarkable thing: the answer does not depend on the radius of the circle! This means we have solved the problem for any size sphere (or one might say for every planet or spherically shaped object).

Hence, for Jupiter, rope D is also longer than C by about 6.28 feet.

I remember doing something like this in high school, but we certainly used $1 bills instead of $100 bills. These guys had more confidence than we did back then. Check it out:

Video: burning a $100 bill

The key to the whole experiment is having an alcohol-water mixture: the alcohol burns and the water absorbs the heat. There are more details in this article.

Here are the results:

Of the 75 people who answered, a majority replied giving $200 or less on a single gift.

The rest gave some pretty nice gifts, and 1 in 5 said they spent more than $1,000.

In hindsight, I realize that I set up really bad categories for the poll. What I should have asked is how much people were planning to spend for holiday gifts.

The American Research Group does an annual survey of gift giving. In 2011, it found that people planned to spend about $650 in gifts. (via: frugal zeitgeist)

Spending $1,000 or more on a gift is a lot, but perhaps not that surprising. After all, there are many parents who would exceed spending $1,000 in a gift to a child, in the form of something like education/a car/downpayment on a house/a laptop.

Thanks to everyone that answered this poll question!