## What’s in the Name? Solution

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.
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