## Archive for November 11, 2010

### Joy in Repetition

Today is 11/11, a day of repeated digits, which makes it a good day to share an email with you that I recently received. My coworker Julia blows me out of the water when it comes to being a number geek, and that’s saying something. Of course, her love of mathematics may be destiny — her name is Julia, and her brother’s name is Vitali. This is the message she sent me:

This Thursday, my dad will be 55 years old. On that same day, I’ll be 11,111 days old. It’s a day of repeated digits for [my] family.

I don’t know what compelled her to do the calculations that allowed her to figure that out, but it reminded me of some repdigit problems. I’ll share those in a minute. But first, a question about Julia:

How old was Julia’s dad when she was born?

Okay, that one was pretty easy. Here’s my favorite repdigit problem, which is a little tougher but can still be attempted by most anyone:

What is the smallest positive integer that, when multiplied by 7, gives a positive integer result in which every digit is a 5?

Of course, there’s the really cool repdigit number pattern:

1 × 1 = 1
11 × 11 = 121
111 × 111 = 12,321
1,111 × 1,111 = 1,234,321

What is the value of the product 1,111,111,111 × 1,111,111,111?

And to end all this silliness, just some facts about repdigit polygonal numbers — numbers that repeat the same digit that are also polygonal numbers. Define P(k,n) to be the nth polygonal number with k objects on a side. For instance, P(3,4) = 10, because P(3,4) is the notation for the 4th triangular number (k = 3). Then P(5,4) = 22 is a repdigit polygonal number, and so is P(8,925,662,618,878,671; 387) = 666,666,666,666,666,666,666. Wow.

As it turns out, there’s a formula for these beasts:

P(k,n) = (n/2)(k – 2)(n – 1) + n  (for n, k > 1)

Over and over,
she said the words
’til he could take no more…

– Prince, Joy in Repetition

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

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