24

720

111100

and

drumroll please....

And the winner is.....well, that's not so clear. We'll discuss it below. We'll also tell you what evil numbers and abundant numbers are.

But, first, a few important words of thanks!

We had a GREAT Middle School Math Meet today! Thanks very much to Felix Sun, Qun Lu, and the Principal of the CCC Chinese School Jianzhong Tang for making arrangements to host our December Middle School Math Meet at Shaker Junior High this afternoon. Thanks as well to Felix's mother, Le Xu, for organizing refreshments!

Thanks to UAlbany Professor Rita Biswas, Hebrew Academy math teacher Alexandra Schmidt, and Doyle Middle School math teacher Nancy Smith for helping to run the Math Meet. Thanks as well to our outstanding high school student coaches: Felix Sun (Shenendahoah High School), Zubin Mukerjee (Guilderland HS), Cecilia Holodak and Flora Mao (Niskayuna HS), Simran Rastogi and Gili Rusak (Shaker).

Thanks to all the students who came and worked enthusiastically on the problems.

Thanks to George Reuter of mathmeets.com, who did a great job of writing more great contest problems for the December math meet. We can't discuss those questions yet, since other teams may still be taking that contest.

Okay, so back to this evil and abundant question, which we CAN discuss, since I just created it as a little supplementary challenge to fill in the bits and pieces of waiting time during the meet. It turned out to be way more interesting than I had realized!

We began the Math Meet by discussing the "number of the day: 12." (Why, because it is December 12, or 12/12, of course!)

Like all numbers, 12 has many interesting properties. We focused on two of those properties today, which are highlighted in the Tagxedo-produced graphic above.

Twelve is an "evil number", which means that it has an even number of ones in its binary expansion, i.e., 1100

_{base 2}= 1*8 + 1*4 + 0*2 + 0*1 = 12.

Twelve is also an "abundant number," because 12 is less than the sum of its proper factors, i.e., 12 < 14 = 1+2+3+4+6. In fact, it is the smallest abundant number, and therefore, of course, it is also the smallest evil abundant number, as well.

Is there a largest evil abundant number?

Some students noted that doubling an evil number always gives you another evil number! (Why?) What if you tripled an evil number? Or multiplied your evil number by other integers? What if you add two evil numbers? What if you raise an evil number to a power? Do you always get another evil number?

(By the way, you could ask the same questions about "odious numbers," which are the opposite of evil numbers--they have an odd number of ones in their binary expansions.)

Other students noted that doubling an abundant number always gives you another abundant number! (Why?) What if you tripled an abundant number? Or multiplied your abundant number by other integers?

Again, you could ask the same questions about perfect numbers, or deficient numbers.

More interesting questions: can a power of two ever be an evil number? Why or why not? Can a power or two ever be an abundant number? What about powers of three?

All great questions to think about!

Now back to judging the entries submitted in the contest.

24 is clearly evil (binary representation is 11000) and abundant (its proper factors are 1,2,3,4,6,8,12, which sum to more than 24.)

720 is also evil (binary representation is 1011010000) and abundant (its proper factors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360).

111100 turns out to be trickier. It is not clear whether it was intended to be interpreted as a binary number, in which case it would obviously be evil. If it was intended to be interpreted as a base-10 number, then it's actually not evil, because the binary representation of 111100

_{base 10}is 11011000111111100. So the only way this number is a candidate is if we interpret 111100 as intended to be binary, in which case it is only equivalent to decimal 60. Now, 60 is clearly abundant, but it is less than 720.

Now, for that humongous number with all the concatenated exponents. I can tell you it is abundant, for sure, but I have no idea if it is evil or not. I am not convinced that the team that submitted it is sure whether it is evil or not, but if they can come up with a convincing proof that it is evil, I am willing to listen. Thinking systematically about some of the questions I raised above may help you sort out what is going on with your--very interesting--number. You may also want to consult the following book, which introduced the concepts of evil and odious numbers: Winning Ways for Your Mathematical Plays Volume 3, by Elwyn R. Berlekamp, John Horton Conway, and Richard K. Guy.

In the meantime, until such time as the orange team can demonstrate their candidate number is evil, 720 is the winner among the numbers submitted today.

Congratulations to the Green Team: Sean, Jason, Gideon, Frank, and Aaron. Your entry of 720 is the largest confirmed evil abundant number of those submitted today.

And, everyone, keep on thinking about this problem! Can you come up with a formula for an arbitrarily large evil abundant number that you can PROVE will work?

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