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
Which leads to the question:
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