Number Gossip on Steroids

I’ve been too busy lately, so I stopped checking my Number Gossip website. Luckily, my website has fans. So one of them, Neil, notified me that my website was hijacked, and instead of describing properties of numbers, was selling steroids. I emailed Dreamhost, my hosting provider. They requested proof that I owned the domain. Why didn’t they request proof from the people selling steroids? Or were they selling steroids themselves?

I fixed my steroid issue and since I was thinking about it anyway, I decided to update Number Gossip. I ended up spending tons of time on it — I had ten years of emails with suggestions for new properties, and I went through all of them and added the interesting ones. For example, Joshua Gray emailed me a cute property of 1331 mentioned on Wikipedia: 1331 was said to be the only cube of the form x2 + x − 1. I didn’t see how to prove it, so I posted it as a question on mathoverflow. It turns out that 1331 is actually not the only cube of this form. There are three of them: −1 (for x = 0 or −1), 1 (for x = 1 or −2), and 1331 (for x = 36 or −37). So 1331 is the only non-trivial cube with this property. I had to fix Wikipedia. By the way, did you notice a symmetry? Plugging in x and −x − 1 into the quadratic produces the same value.

After processing all the emails related to Number Gossip, I got excited, so I continued working on it and added tons of new unique properties. Some of them I invented myself, some more were inspired by sequences in the OEIS database. I now have a collection of my new favorite unique properties, which I will post soon.

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Continue the Sequence: 742, …

This is the sequence of numbers n such that 3 times the reversal of n plus 1 is the number itself. In other words, n = 3*reversal(n)+1. For example, 742 = 3*247+1. In fact, 742 is the smallest number with this property. How does this sequence continue, and why?

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I am Hooked on Star Battles

I recently published Sergei Bernstein’s awesome Star Battle called Swiss Cheese. Another lovely Star Battle from him is called Hooks. You can play it online at puzz.link.

Sergei's Star Battle

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Swiss Cheese Star Battle

Star Battle is one of my favorite puzzle types. The rules are simple: put two stars in each row, column, and bold region (one star per cell). In addition, stars cannot be neighbors, even diagonally.

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.

Swiss Cheese Star Battle

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


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Our Familect Story

A familect is a portmanteau word formed by squashing together two words: family and dialect. It means words that a family uses that are not a part of a standard vocabulary. This story is about two words that my son Sergei invented, and twenty years later, my family still uses them.

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.

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A Splashy Math Problem

A problem from the 2021 Moscow Math Olympiad went viral on Russian math channels. The author is Dmitry Krekov.

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 An by 2?

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I Miss John Conway

Yesterday I stumbled upon a picture of John Conway I completely forgot I had: it was saved in a wrong folder. The photo was taken at the MOVES conference in 2015. There is a third person in the original shot, but I do not remember her. I decided to cut her out as I didn’t know how to contact her for permission to post this photo.

Conway at MOVES conference 2015.

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The Anniversary Coin

Konstantin Knop, the world’s top authority on coin-weighing puzzles, suggested the following problem for the 2019 Russian Math Olympiad.

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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Discover the Rule

Found the following cute puzzle on Facebook.

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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Build an All-red Cube

This is one of my favorite problems given at the 2017 Moscow Olympiad to grades 6 and 7. It was suggested by one of my favorite problem writers: Alexander Shapovalov.

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.

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