# 2 More Pickover Puzzles

Every now and then Pickover tosses out an extremely obvious puzzle. Given the following sequence of numbers:

```  1, 27, 125, 343, 729, 1331, 2197, 3375, x
```

what is x? Now, when I say “obvious” in this context I should clarify that the solution is relatively straightforward to the math-trained mind. Typically the solutions to such numerical puzzles are along the lines of “The latitude coordinates of the original U.S. colonies multiplied by the longitude coordinates of the same, then add 3.”

Anyway, that was today’s puzzle. The puzzle for April 4 was to find the numbers, A and B, that satisfy the following equation:

```  AB + BA = A7
```

My first impulse was there should be some way to solve for A and B, maybe involving logarithms or something, until I noticed that there were 2 variables and only 1 equation which, per my math classes of long ago, proves difficult. So, a brute force program it was. Surprisingly, there is only 1 solution.

See [more] for the answers, I mean, if you really care.

I have a stack of puzzles to work through that should be brute-force-able using a combination of permutation logic and graph techniques. Eventually, I imagine I will be able to assemble a program toolkit to rip through an entire year of these puzzles automatically.

# Brute Force Word Puzzle

I used to think that it would be too far-fetched to write a computer program to determine the solution(s) to one of Pickover’s more linguistic puzzles, such as the puzzle for 2006/03/22; find letters to fill in positions 1, 3, and 5 to form an actual word:

```1 n 3 i 5 n
```

I decided to call in the help of a program I had only heard of — Ispell — that I previously had no use for because I’m the most awesome spehlurr I personally know. However, I researched it, installed it, and determined that I could use it to come up with a list of plausible puzzle solutions.

# Big Number Puzzle

Big Brain’s puzzle for the weekend of March 11-12 was essentially an anagram puzzle. Using the following 12 letters:

e c m a o n s a d e l k

form an 8-letter (English) word and a 4-letter (English) word. One word tastes good and the other carries stuff. Usually, I am at a loss to write a program that can re-sort letters and find what look to be valid words. However, I recently found an anagram generator. Go to the homepage of the GNU MP Bignum library, scroll way to the bottom and find “Demo 2 — Find Anagrams”. One day, I hope to understand how big numbers pertain to computing anagrams. Anyway, insert those 12 letters and you will find an impressive list of anagrams, including the correct answer for the puzzle! For added enjoyment, insert your name or those of your friends (the Guru’s name anagrams to “Hemimeridae larceny”).

Your reign of terror-logic will not last long, Pickover. We have technology on our side.

# AAC Puzzles

You thought those Pickover puzzles were inane? Well, you were right. But it does help to keep a programmer sharp on pure computer science skills. One reason I have been having such a good time reverse engineering the FAAD2 source is because it represents a bunch of little puzzles. Only they’re considerably more straightforward and grounded in reality than the Pickover puzzles. If these puzzles were on a Pickover calendar page, they would be worded along the lines of,

“There are 8,192 numbers stored in a table called iq_table. 0, 1, 2.5198420997897464, 4.3267487109222245, all the way up to 165113.4940829452. What is the common property that connects all of these numbers?”

The answer (thanks to Jindrich Makovicka for pointing this out on the ffmpeg-devel list) is that each number is its index (0..8191) raised to an exponent of 4/3: iq_table[index] = index4/3. Here are some more puzzles: