Math Teachers at Play #21 is up at Math Mama Writes…, and it includes this cute puzzle, which Sue apparently made up herself:

The Numberland News runs personal ads. 21 was looking for a new friend and put an ad in.

Two-digit, semi-prime, triangular, Fibonacci number seeks same. I’m a binary palindrome, what about you?Will 21 find a friend?

A *semi-prime* is a number with exactly *two* prime factors, like 6. See this post for a definition of *triangular number*, this post for some hints on how to figure out a general formula for computing triangular numbers, and this one for the solution. Fibonacci numbers are discussed here. Finally, a *palindrome* is a number (or word, or phrase) which is the same forwards and backwards; a *binary palindrome* is a number which is a palindrome when expressed in base two.

I’m glad you like my puzzle. Yes, I did make it up, I’m proud to say. (This is a new skill for me.)

I started out by looking up 21 on Wolfram Alpha, and noting all its properties. I was intrigued by the fact that it was both triangular and Fibonacci, and used a spreadsheet to find others like that. The rest sort of fell into place.

Unless I’m mistaken, 21 will not find a friend but he/she/it will come pretty close with 55. 55 is a two-digit, semi-prime, triangular, Fibonacci number. It’s binary form, 110111, is however not a palindrome. Perhaps 21 can overlook this subtle flaw. Nice problem!

Dave: the way I read it, 21′s “seeking same” didn’t apply to the binary palindrome part. I don’t know what Sue originally had in mind, but to my mind you haven’t

quitefinished solving the problem. If 55 responded to 21′s personal ad, what would it say?I see. In that case, 55 would be a suitable friend. His/Her/It’s response would be, “No, I’m not a binary palindrome.”

55 would say, “Anthropomorphizing numbers is pretty bloody weird. Why don’t you look for companionship in a single digit bar?”

Dave: I think there’s something a little more positive 55 could say… =)

Michael: hah! =)

Well, now I’m stumped!

Dave: 55 isn’t a

binarypalindrome…Oh. So it’s a palindrom in decimal, right?

Dave: yes, and…

Ironically, It hadn’t even occurred to me that 55 could respond that it was a palindrome too (in two bases, as JD put it). ;^)

It’s a palindrome in THREE bases, isn’t it? =)