In the country of Terrachaos, there is a huge number of voters and national elections are held routinely. Each voter is either a Paleomorph or a Neomorph, and each voter gets one vote. There are three political parties: Patriacrats, Plutocrats, and Xenocrats. Every Neomorph always votes for the Xenocrats. Paleomorph voting is more complicated. The percentage of Paleomorphs voting for the Patriacrats is equal to the percentage of voters who are Neomorphs at the time of the election. The remaining Paleomorphs vote at a constant percentage *P* for the Plutocrats and a corresponding fixed percentage 1-*P* for the Xenocrats. For example, if the Neomorphs are 20 percent of all voters at election time and *P* = 70 percent, then the Paleomorphs will vote 20 percent for the Patriacrats, 56 percent for the Plutocrats, and 24 percent for the Xenocrats. Consequently, the overall vote, combining Paleomorph and Neomorph votes, will be: Patriacrats 16 percent, Plutocrats 44.8 percent and Xenocrats 39.2 percent.

If any party gets 50 percent or more of the overall vote, that party forms the government and implements its policies. If no party gets at least 50 percent, then the Plutocrats form a coalition government with the other party that got the most votes, and the policies of that other party are implemented. In the previous example, the Plutocrats would partner with the Xenocrats and implement Xenocrat policies, even though the Plutocrats got more of the total votes.

Under the policies of either the Plutocrats or the Xenocrats, the ratio of Neomorphs to Paleomorphs will increase by 10 percent by the time of the next election. For example, if there are 10 Neomorphs for every 100 Paleomorphs at the time of an election, these policies will lead to 11 Neomorphs for every 100 Paleomorphs at the time of the next election. The policies of the Patriacrats will keep the ratio of Neomorphs to Paleomorphs constant until the next election, or 10 Neomorphs for every 100 Paleomorphs in that election.

The Neomorphs are initially a small, but non-zero, fraction of the voters. Depending on the value of P, as described previously, what will happen over time to the Neomorphs as a percentage of the voters and with respect to the political party composition of the government?

Alternatively, suppose the Paleomorphs vote for the Patriacrats in a percentage equal to the square root of the percentage of voters that are Neomorphs at election time. For example, if 4 percent of the voters are Neomorphs, then the Paleomorphs vote 20 percent for the Patriacrats. Otherwise, everything is the same as previously described. What happens over time?

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## Stealthy Cruise Missiles

In this puzzle, a non-stealth cruise missile flying at 2,000 miles is detected by the radar at its target when it is 1,000 miles away, giving 30 minutes’ warning. A somewhat stealthy cruise missile, of similar physical shape and size, diffusely reflects the radar energy at a rate that is only 10 percent the rate of the non-stealth missiles, but only flies at 1,000 miles per hour. The first question is: What is the warning time for this missile? An advanced stealth missile, flying at only 500 miles per hour, is designed to give only 5 minutes’ warning. The second question is: What is the rate of radar energy reflection for this missile relative to the non-stealth missile?

The key to this puzzle is the “radar equation” which implies that the energy *E* received back at an active radar station is, all other relevant things being equal, inversely proportional to the 4^{th} power of the distance *r* of the target from the station. So, the minimum energy threshold for radar detection of the somewhat stealthy missile will occur when 10% *r** ^{–}*

^{4}≥ (1,000 miles)

^{-4}, or when

*r*≤ 562 miles. Since this missile flies at 1,000 miles per hour, that will give about 33 minutes and 45 seconds of warning time.

To be detected only five minutes before it hits, the stealthy missile will need to be first detected when it is 41.67 miles away. So, its relative rate of radar reflection *a* will need to satisfy *a*(41.67 miles)^{-4} ≥ (1,000 miles)^{-4} or *a* ≤ 0.0003 percent.

Solutions were also submitted by Bob Conger, Dominick Elia, Kristen Fox-Neff, Clive Keatinge, Matthew Pear and David Zheng.