A Plate of 1,000 Cookies

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Image: Wikimedia Commons

A puzzle by David B., a mathematician at the National Security Agency, from the agency’s May 2017 Puzzle Periodical:

Steve, Tony, and Bruce have a plate of 1,000 cookies to share. They decide to share them in the following way: beginning with Steve, each of them in turn takes as many cookies as he likes (they must take an integer amount, greater than or equal to 1), and then passes the plate clockwise (with Tony sitting to Steve’s left, and Bruce sitting to Tony’s left). Nobody wants to feel like he hogged too many cookies, so they all want to avoid being the player at the end who has taken the most cookies. Additionally, nobody wants to feel cheated by finishing with the fewest cookies. Finally, given that the previous two conditions are definitely met, or definitely cannot be met, each player would like to maximize the number of cookies he eats. The players’ objectives can be summarized as follows:

Objectives:

  1. Have one player who has eaten more cookies than you, and one player who has eaten fewer cookies than you.
  2. Eat as many cookies as possible.

Objective #1 takes infinite priority over Objective #2. Assuming that all players are perfectly rational, that they are all aware of each other’s rationality and objectives, and that they cannot communicate with each other in any way, how many cookies should Steve take to ensure he meets both objectives and how many cookies will Tony and Bruce take if Steve takes the winning amount?

Click for Answer
| Puzzles

Extended Engagement

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Image: Wikimedia Commons

The upper edge of the setting sun is sometimes seen to take on a green tinge, an effect of atmospheric refraction. Normally this is apparent only briefly, but for Richard Byrd’s Antarctic expedition of 1928-1930 it lasted more than half an hour:

Here the sun descends so slowly that it seems to roll along the horizon and as it will be only two days until it is above the horizon all the time for the rest of the summer it clings interminably before, with seeming reluctance, dropping from sight. As its downward movement is so prolonged the last rays shimmer above the barrier edge as it moves eastward, appearing and reappearing from behind the irregularities of the barrier surface. It trembles and pulsates, producing a vibration light of great beauty.

The night the green flash was seen some one ran into the administration building and called, ‘Come out and see the green sun.’

There was a rush for the surface and as eyes turned southward, they saw a tiny but brilliant green spot where the last ray of the upper limb of the sun hung on the skyline. It lasted an appreciable length of time, several seconds at least, and no sooner disappeared than it flashed forth again. Altogether it remained on the horizon with short interruptions for thirty-five minutes.

When it disappeared momentarily it seemed to have been shut off by a tiny spurt, an inequality in the skyline caused by the barrier surface.

“Even by moving the head up a few inches it would disappear and reappear again and after it had finally disappeared from view it could be recaptured by climbing up the first few steps of the [antenna] post.”

(From an account by witness Russell Owen, San Francisco Chronicle, Oct. 23, 1929.)

| Science & Math

Public Spirit

The Guinness record for the most fraudulent election ever reported belongs to the Liberian general election of 1927, in which President Charles D.B. King was re-elected over challenger Thomas J. Faulkner:

Candidate Votes %
Charles D.B. King 243,000 96.43
Thomas J. Faulkner 9,000 3.57
Total 252,000 100.00

As there were fewer than 15,000 registered voters, this represents a turnout of 1,680 percent — robust indeed.

| Society

Evolution

I just ran across this anecdote by Jason Rosenhouse in Notices of the American Mathematical Society. In a middle-school algebra class Rosenhouse’s brother was given this problem:

There are some horses and chickens in a barn, fifty animals in all. Horses have four legs while chickens have two. If there are 130 legs in the barn, then how many horses and how many chickens are there?

The normal solution is straightforward, but Rosenhouse’s brother found an alternative that’s even easier: “You just tell the horses to stand on their hind legs. Now there are fifty animals each with two legs on the ground, accounting for one hundred legs. That means there are thirty legs in the air. Since every horse has two legs in the air, we find that there are fifteen horses, and therefore thirty-five chickens.”

(Jason Rosenhouse, “Book Review: Bicycle or Unicycle?: A Collection of Intriguing Mathematical Puzzles,” Notices of the American Mathematical Society, 67:9 [October 2020], 1382-1385.)

| Puzzles · Science & Math

Bottema’s Theorem

Grab point B above and drag it to a new location. Surprisingly, M, the midpoint of RS, doesn’t move.

This works for any triangle — draw squares on two of its sides, note their common vertex, and draw a line that connects the vertices of the respective squares that lie opposite that point. Now changing the location of the common vertex does not change the location of the midpoint of the line.

It was discovered by Dutch mathematician Oene Bottema.

(Demonstration by Jay Warendorff.)

| Science & Math

The Aventine Keyhole

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The keyhole of the Priory of the Knights of Malta in Rome presents a perfectly framed view of Saint Peter’s Basilica.

It’s not clear whether this is a happy accident or a deliberate design. The property lies in the piazza Cavalieri di Malta, which was designed in 1765 by the supremely imaginative Giovanni Battista Piranesi — who imagined the Aventine Hill as a sacred ship that would sail to the heavens.

| Art · History · Oddities