Tanya Khovanova's Math Blog

I recently posted my puzzle designed for the MoMath meet-up.

What’s in the Name?

• 4, 6, X, 9, 10, 12, 14, 15, 16
• 1, 2, 6, 24, 120, X, 5040, 40320, 362880
• 2, X, 3, 4, 7
• 1, 2, 3, 4, 5, 6, X, 8, 9, 153, 370, 371
• X, 2, 3, 4, 5, 6, 7
• 6, 28, 496, 8128, X, 8589869056, 137438691328
• 0, 1, 1, X, 4, 7, 13, 24, 44, 81

Now it is time for the solution.

The solvers might recognize some sequences and numbers. For example, numbers 6, 28, and 496 are famous perfect numbers. Otherwise, the solvers are expected to Google the numbers and the pieces of the sequences with or without X. The best resource for finding the sequences is the Online Encyclopedia of Integer Sequence.

The first “AHA!” happens when the solvers notice that the sequences’ names are in alphabetical order. The order serves as a confirmation of the correctness of the names. It also helps in figuring out the rest of the sequences’ names. The alphabetical order in such types of puzzles hints that the real order is hidden somewhere else. It also emphasizes that the names might be important. The sequences names in order are:

• Composite
• Factorial
• Lucas
• Narcissistic
• Natural
• Perfect
• Tribonacci

The second “AHA!” moment happens when the solvers realize that the Xs all have different indices. The indices serve as the final order, which in this case is the following:

• Natural
• Lucas
• Composite
• Tribonacci
• Perfect
• Factorial
• Narcissistic

The third “AHA!” moment happens when the solvers realize that the number of terms is different in different sequences. It would have been easy to make the number of terms the same. This means that the number of terms has some significance. In fact, the number of terms in each sequence matches the length of the name of the sequence. The solvers then can pick the letter from each of the names corresponding to X. When placed in order, the answer reads: NUMBERS.

The answer is related to the puzzle in two ways:

1. The puzzle is about numbers.
2. The sequences’ names do actually need the second word: Lucas numbers, composite numbers, and so on.

The advantage of this puzzle for zoomed group events is that the big part of the job — figuring out the sequences — is parallelizable. Additionally, it has three “AHA!” moments, which means different people can contribute to a breakthrough. The puzzle also has some redundancy in it:

1. Due to the redundancy of the English language, it is possible to solve this puzzle without figuring out the names of all the sequences.
2. If the solvers can’t figure out the order, they can anagram the letters to get to the answer.
3. If the solvers do not realize that they have to use the letter indexed by the X, there is another way to see the answer: read the diagonal when the sequences’ names are in order.

How can a square be square-free? In order for square-freeness to be interesting, it must be, and is, defined in terms of divisibility by non-trivial squares. So to create this particular mathematical oxymoron, one just needs a trivial square, namely 1.

I collect exciting properties of integers on my Number Gossip website. Did you know that forty is the only number whose letters appear in alphabetical order when written in English? Or that the largest amount of money one can have in US coins without being able to make change for a dollar is 119 cents?

Recently I wrote about a weird occasion that motivated me to search for new properties. Here is a sample of some amusing new updates.

• 68 is the last 2-digit string to appear in the decimal expansion of pi.
• 77 is the smallest number n such that the smallest possible number of multiplications required to compute x to the n-th power is by 1 fewer than the number of multiplications obtained by Knuth’s power tree method.
• 195 is the smallest groupless number; that is, it is the smallest number n whose set of pairwise products up to n cannot be completed to the multiplication table of a finite group of order n (submitted by Andrew Pollington).
• Digits in the Morse code can be represented in base 3 with 1 for a dot and 2 for a dash; Morse-coded zero in base 3 is evaluated to 242.
• 247 is the smallest possible difference between two integers that together contain each digit exactly once.
• An equilateral triangle, whose area and perimeter are equal, has an area (and perimeter) equal to the square root of 432.
• 480 is the smallest number such that when written in hexadecimal (1E0), looks like another number in scientific E notation.
• 510 is the smallest number that is not a palindrome, even after removing trailing zeros, which is divisible by its reversal. It is trivial that palindromes with trailing zeros are divisible by their reversal, making this the first interesting case.
• 960 is the number of different starting positions in the Fischer random chess game. The Fischer random chess was invented, not surprisingly, by Bobby Fischer. The starting position is created like this: White’s non-pawn pieces are placed randomly on the first row so that the two bishops are on different colored squares, and the king is between the rooks to allow to castle. The white pawns are placed on the second row as usual, and the black pieces are placed symmetrically with respect to the horizontal line symmetry.
• 1000 is the smallest number whose scientific notation (1E3) is shorter than its decimal representation.
• 1084 is the smallest number whose American English name contains all five vowels in order.
• 1642 is the smallest number n without zeros so that for each digit d of n, the number 2^d is a substring of n.
• 1659 is the smallest number n such that its Roman numeral occupies the n-th position when all Roman numerals are sorted in the lexicographic order.
• 2187 is the smallest compact number: numbers that can be expressed more compactly using their prime factorization than their decimal expansion, where multiplication and power contribute one character (2187 is written as 3^7).
• 2500 is the smallest number n whose distinct prime factors are exactly the same as the distinct prime factors of each of the numbers obtained by deleting any single digit of n.
• 8542 is the largest integer that can’t be represented as a sum of squares of numbers whose reciprocals sum to 1. This is a recent (2018) result by Max Alekseyev.
• 8719 is the smallest number n such that the smallest possible number of multiplications required to compute x to the n-th power is by 2 fewer than the number of multiplications obtained by Knuth’s power tree method.

• 4, 6, X, 9, 10, 12, 14, 15, 16
• 1, 2, 6, 24, 120, X, 5040, 40320, 362880
• 2, X, 3, 4, 7
• 1, 2, 3, 4, 5, 6, X, 8, 9, 153, 370, 371
• X, 2, 3, 4, 5, 6, 7
• 6, 28, 496, 8128, X, 8589869056, 137438691328
• 0, 1, 1, X, 4, 7, 13, 24, 44, 81

This is the puzzle I designed for yesterday’s event at the Museum of Mathematics. This puzzle is without instructions — figuring out what needs to be done is part of the fun. Solvers are allowed to use the Internet and any available tools. The answer to this puzzle is a word.

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.