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My son, Sergei Bernstein, recently designed a Star Battle with a beautiful solve path. This is my favorite Star Battle so far. I like its title too: Swiss Cheese.

You can also solve it at the puzz.link Star Battle player.

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As you might know, I was married three times, and I have two sons, from two different fathers. These fathers were also married several times and had other children. My two sons are half-brothers, and they also have half-siblings through their fathers. Thus a half-sister of a half-brother became a *quarter-sister*. Inventing this term was quite logical for a son of two mathematicians.

As you can imagine, our family tree is complicated. One day Sergei pointed out that our tree doesn’t look like a standard tree and called it a *family bush*.

**Problem.** Does there exist a number *A* so that for any natural number *n*, there exists a square of a natural number that differs from the ceiling of *A*^{n} by 2?

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Puzzle.Eight out of sixteen coins are heavier than the rest and weigh 11 grams each. The other eight coins weigh 10 grams each. We do not know which coin is which, but one coin is conspicuously marked as an “Anniversary” coin. Can you figure out whether the Anniversary coin is heavier or lighter using a balance scale at most three times?

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Puzzle.Discover the rule governing the following sequence to find the next term of the sequence: 8, 3, 4, 9, 3, 9, 8, 2, 4, 3.

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Problem.We are given eight unit cubes. The third of the total number of their faces are blue, and the rest are red. We build a large cube out of these cubes so that exactly the third of the unit cube’s visible faces are red. Prove that you can use these cubes to build a large cube whose faces are entirely red.

With my PRIMES student, Sean Li, we looked at this game and asked a different question. Suppose Alice picks a pattern of three tosses in a row that are the same. Suppose after that, Bob chooses a pattern of three alternating tosses. Then they toss a fair coin. Alice is hoping for HHH or TTT, while Bob is hoping for HTH or THT. The person whose pattern shows up first wins. For example, if the tosses are THTTHHH, then Bob wins after the third toss. For these particular choices, Bob wins with probability 1/2.

In this example, what actually happens. We assume that the group of two elements acts on the alphabet of two letters. The group’s non-identity element swaps letters H and T. We assume that two strings are equivalent if they belong to the same equivalency class under the group action. We call such an equivalency class a pattern.

In the new game we invented, we have an alphabet of any size and any group acting on the alphabet. Then Alice and Bob pick their patterns. After that, they play the Penney’s game on these patterns. The answers to all the relevant questions are in our paper, *The Penney’s Game with Group Action,* posted at the math.CO arxiv 2009.06080.