## Archive for May 6, 2011

### Wu, California Earn MathCounts Titles

Today, I had the privilege of attending the MathCounts National Competition. I served as a volunteer in the scoring room, and I had the pleasure of scoring the competitions of the winning team. Congratulations to the California team of Celine Liang, Sean Shi, Andrew He and Alex Hong, who scored an amazing 59.5 (out of 66) on the competition.

You may be thinking that 90.1% correct is not very impressive. With the current level of grade inflation, I suppose that’s a reasonable reaction. But given that these students solve 48 non‑trivial questions (see examples below) in under two hours, it’s an amazing feat. Given twice as much time, most of us would be lucky to answer half of them correctly.

During the Target Round of the competition, students are given six minutes to solve a pair of problems. My favorite problem on this year’s competition was the last question in the Target Round:

How many positive integers less than 2011 cannot be expressed as the difference of the squares of two positive integers?

It’s a good question. You might like to try solving it. But don’t be too discouraged if it takes you more than six minutes… after all, no one expects you to be as smart as a middle school student.

One of the California competitors, Sean Shi, scored well enough in the written competition to qualify for the Countdown Round. The Countdown Round is an NCAA‑style bracket for the top twelve competitors. All students who qualify for the Countdown Round are introduced to the audience, and the following story was told about Sean:

Sean’s younger brother is two years younger than Sean. When Sean was 7 and his brother was 5, his brother told their mom, “You’re the best mommy in the whole world!” Already a mathematical prodigy, Sean replied, “Actually, the chance of that being true is very low.” Sean explained, “No offense, mom! I’m just being mathematical!”

Another competitor in today’s Countdown Round, Shyam Narayanan of Kansas, finished among the top four in last year’s national competition. As a result, he was invited to meet President Obama in an Oval Office ceremony. During the visit, Shyam asked the President a question that he said he had never been asked before.

Since the Oval Office is an ellipse, where are its foci?

Although Obama didn’t know the answer, the mathletes in attendance were able to help him figure it out.

Though Shi and Narayanan made respectable showings in the Countdown Round, it was Scott Wu from Louisiana who prevailed and earned the title of national champion. Wu, who’s older brother Neil Wu was the 2005 MathCounts National Champion, defeated Yang Liu 4‑1 in the finals. Wu locked up the title by correctly solving a rather odd question about digits.

It takes 180 digits to write all of the two-digit positive integers. How many of the digits are odd?

In the Countdown Round, students only have 45 seconds to answer each question — though typically, they answer the questions much faster. So how do you come up with an answer to this question in just a few seconds?

First, realize that exactly half of the two‑digit numbers have an odd units digit. Then, notice that 50 numbers have an odd tens digit, but only 40 have an even tens digit. Consequently, of the 180 digits required, there will be 10 more odd digits than even. Hence, 95 digits are odd, and 85 digits are even.

### The Anagram Game

I have a question.

Is AEGNNTT an anagram of TANGENT?

According to wordsmith.org, it’s not. That site defines an *anagram* as “a word or phrase formed by rearranging the letters of another word or phrase.”

But the game *Anagrams* involves forming words from a random collection of letters.

In the book Word Freak, author Stefan Fatsis calls AEGNNTT an *alphagram* of TANGENT. It’s a rearrangement of the letters in alphabetical order.

Recently, my sons and I have been playing The Anagram Game. Generally, I give them a random collection of letters, so perhaps the name is a misnomer. In any case, they have fun trying to rearrange the letters into real words. Sometimes, I present them with a collection of letters that happens to form a word; for instance, tonight I gave them N-I-G-H-T. They quickly rearranged the letters to form THING. Sometimes, I put the letters into alphabetical order, like A-A-B-B-E-L-L-S. Eli impressed me by decoding that one in less than a second: BASEBALL. But most times, I give them a random arrangement of letters, like U-N-A-T, which can form TUNA or AUNT.

Anyone know what you call an random arrangement of letters that can be rearranged to form a word or phrase? If no such word exists, then I would like to suggest *scramblagram*.

While playing the Anagram Game yesterday, I presented Alex with G-O-N-L. “No, daddy,” he said, “the best anagram of LONG is N-G-L-O.”

Well. I guess I stand corrected.

I asked him, “Is there a best anagram of LAWN?”

He immediately answered, “W-N-L-A.”

“And is there a best anagram of WALK?”

He again answered quickly: “L-K-W-A.”

Interestingly, Eli agreed with him in every case. Apparently my four-year-old sons are privy to a universal truth regarding the reordering of letters in a word, about which I am helplessly unaware.

Luckily, Alex was willing to reveal the pattern. “I always put the two back letters first, then the first letter, and then the vowel.”

So there you have it. The system only works for words of the form consonant-vowel-consonant-consonant, but that’ll do just fine if you’re looking for scramblagrams of LIST, PARK, SULK, BOWL, or RENT.

As it turns out, the word *anagrams* itself has an anagram: *ars magna* (Latin, “great art”).

The following are several of my favorite mathy anagrams.

ELEVEN + TWO = TWELVE + ONE

decimal point = I’m a dot in place

schoolmaster = the classroom

What is the square root of nine? = THREE, for an equation shows it!