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.
The Math Less Traveled 
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 quite finished 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 binary palindrome…
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? =)