## Variations on a Theme

Three variations of one of my favorite puzzles. The first is silly; the second is doable; and, the third will take a little bit of jiggering. I don’t know where I first saw this puzzle, but I’m pretty sure the version with ten blanks is in Gödel, Escher, Bach.

Instructions: Place numerals in the blanks to make the sentence true.

This version is for little kids. Or is it?

There are __ zeroes and __ ones in this sentence.

I’m fairly certain there are no solutions when this is extended to three blanks, but four blanks will work:

There are __ zeroes, __ ones, __ twos, and __ threes in this sentence.

It works with seven blanks (when the greatest digit is six), but that’s not much different than the one above. The piece de la resistance is the one with ten blanks:

There are __ zeroes, __ ones, __ twos, __ threes, __ fours, __ fives, __ sixes, __ sevens, __ eights, and __ nines in this sentence.

Have fun!

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• 1. puntomaupunto  |  January 10, 2014 at 4:15 am

Is “There are no zeroes and no ones in this sentence” acceptable? Otherwise, I’d stick to “There are 102 zeroes and 102 ones in this sentence” (subscripts mean that numbers are written in base 2)

• 2. venneblock  |  January 10, 2014 at 8:56 pm

The instructions ask for numerals in the blanks, and “no” isn’t a numeral. And I hadn’t considered your base-2 solution, but I love it. Honestly, I had intended for the first one to be unsolvable — but I should have known folks who read MJ4MF would find a solution! Nice work.

• 3. Chris Smith  |  January 10, 2014 at 10:41 am

*spoiler*
I thought 1,3,1,3 worked for the second but not sure how that relates to today specifically! What have I missed-another solution?Some funny date connection? Help me out :)

• 4. Chris Smith  |  January 10, 2014 at 11:33 am

And 1,2,3,2 also works! Still not sure about the connection though Patrick!!

• 5. venneblock  |  January 10, 2014 at 8:51 pm

Chris, I originally intended to post this on a different date… then I moved it, but I forgot to remove the hint about the date. D’oh! I won’t tell you what date I had originally planned to post it, though — that date is a solution to the puzzle.

FYI, I don’t think either of your solutions is correct. Your first solution says there are three 1’s and three 3’s, and there are only two of each; and your second solution says there are three 2’s, but there are only two.

For those who don’t understand Chris’s message or this reply… this post originally contained the following, in reference to the second puzzle:

hint: I posted this puzzle today because of a solution to this version

• 6. xander  |  January 10, 2014 at 7:58 pm

Assuming that the words “zero[s],” “one[s]“, and so on count as zeros, ones, and so on, I have solutions for the first two, but the last is a doozy. I’ll have to think about that this weekend. Thanks. :P

• 7. venneblock  |  January 10, 2014 at 8:53 pm

I hadn’t intended for “zero” to count as a 0, but if you do, I suppose that’s fine. There are different solutions to the last puzzle if you choose to count them or not.

That’s what I like about this puzzle, Xander — solutions to the first two can be found by trial-and-error, but the last one needs a slightly more robust strategy (which I’ll share in a few days).

• 8. Chris Smith  |  January 11, 2014 at 10:26 am

Patrick, I’d assumed that the words counted towards the total count…in which case these worked:

There are 1 0s, 3 1s, 1 2s and 3 3s in this sentence
and
There are 1 0s, 2 1s, 3 2s and 2 3s in this sentence

If you DON’T count them then I guess:

There are 1 0s, 2 1s, 1 2s, 0 3s in this sentence would be one way to do it…

Thanks for sharing it!

• 9. venneblock  |  January 11, 2014 at 10:44 am

I meant to not count them, which is why I spelled them out. But whatever… you found solutions that worked given your assumptions, and that’s good enough for me!

And 1, 2, 1, 0 was the solution I had expected, and I had planned to post this on 12/10.

Sent from my Verizon Wireless 4G LTE DROID

• 10. Chris Smith  |  January 11, 2014 at 10:57 am

Ha ha…Good stuff.

Hope the Vennebush clan had a great Christmas and are ready for lots of fun in 2014!

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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