## Posts tagged ‘digits’

### 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!

### A Great Day for a Math Trick

Today is 7/11/13, and boy, have I got a great math trick for today! You’ll likely need a calculator.

- Multiply your age by 12.
- Now add the age of your spouse/brother/sister/friend/uncle/aunt/whomever.
- This should yield a three-digit number. Now, divide by 7.
- Then, divide by 11.
- Then, divide by 13.
- The result should be a number of the form 0.
*abcdef*…, with a 0 and a decimal point in front of a long string of digits. Add the**first six digits**after the decimal point.

Here’s the cool part. I don’t know your age, nor do I know the age of your spouse, brother, sister, friend, uncle, or aunt. But I do know that after you completed those steps, the result was 27.

Pretty cool, eh?

There are myriad math tricks of this ilk, but this one is my favorite. It’s based on a trick I learned from Art Benjamin, though I think the one above has more panache than his original. Decide for yourself.

- Choose a number from 1 to 70, and then divide it by 7.
- If your total is a whole number (that is, no digits after the decimal point), divide the answer by 7 again.
- Is there a 1 somewhere after the decimal point? I predict that the number after the 1 is 4. Am I right?
- Now add up the first six digits after the decimal point.

Just as with the trick above, the result will always be 27.

Regardless of which trick you prefer, have a happy 7/11! And if you’ve got a few hours to kill, you can try to solve the 7‑11 problem.

### Fishin’ for an April Fools Math Joke

My friend and former boss Jim Rubillo sent me the following email last night:

I am cleaning, and I found this book that you might want:

One Million Random Numbers in Ascending Order. Do you want it, or should I throw it away?

Seemed like an odd book, and I thought he might have gotten the title wrong since RAND Corporation published *A Million Random Digits with 100,000 Normal Deviates* in 1955. (Incidentally, you should visit that book’s page on www.amazon.com, where you’ll find many fantastic reviews, such as, “Once you’ve read it from start to finish, you can go back and read it in a different order, and it will make just as much sense as your original read!” from Bob the Frog, and, “…with so many terrific random digits, it’s a shame they didn’t sort them, to make it easier to find the one you’re looking for,” from A Curious Reader.)

Knowing that Jim is a stats guy, it seemed plausible. A little confused, I wrote back:

Is it literally just a list of random numbers? If so, I’ll pass. But if there’s something more interesting about it, then maybe?

His response?

It’s a sequel to

The Complete Book of Even Primes.

Gotcha!

And so it goes, with April Fools even afflicting the math jokes world.

Thank goodness he didn’t tape a fish to my back.

### Dead Man’s Hand

A few days ago, I posted about the repeating digits that had been occurring in the Amazon sales rank of *Math Jokes 4 Mathy Folks*. For three days in a row, a repeating digit occurred three times in the sales rank each day.

The story continues.

On August 1, it happened again… but something cooler happened, too. The sales rank of *MJ4MF* on August 1 was 88,111. Though sad in regards to book sales, it’s numerically fantastic! Not only does it contain a digit that repeats three times (1), the other digit (8) repeats twice, and — hold on to your hat! — the repeated digits 8 and 1 occurred on the date 8/1!

Does it get any cooler than that?

Well, maybe. It might have been a little cooler if the rank were 8,111, which would match the full date 8/1/11.

But 88,111 could also be thought to represent a “Dead Man’s Hand” in poker. Officially, a Dead Man’s Hand contains a pair of aces and a pair of eights, but many sources also use the term to refer to a full house with aces over eights.

What did the geometrician say when someone took his playing cards?

Decagon.

This reminds me of a poker game that I played with an engineer and a statistician. In the middle of the second hand, the police raided the game. An officer asked the engineer, “Were you gambling?”

The engineer replied, “No, officer, I was not gambling.”

The officer then asked the statistician, “What about you?”

The statistician replied, “No, officer, I was not gambling, either.”

Finally, the officer turned to me. “Were you gambling?” he asked.

I replied, “By myself?”

### 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.

### Can You Digit?

A few folks have asked me about the number characters that appear in the Jokes section of the Math Jokes 4 Mathy Folks website. For those who have no idea what I’m referring to, a number character appears in the left sidebar on each joke page of the site; but you shouldn’t have to jump through hoops to see them, so here is the entire set for your viewing pleasure:

I’m proud to say that those little guys are my original creations.

I learned long ago that the best things in life — such as beer, taxi rides, and tickets to a Justin Bieber concert — are better when shared, so in that spirit, I’d like to share these little guys with you. Below are links to high-resolution images of each of the characters; feel free to download them and use them as clip art, especially if you’re a teacher who’s trying to liven up a boring worksheet (or if you’re just a boring teacher). Clicking on any of the images below will take you to a new page with a large version of the image. On that page, right-click and choose “Save As” to download, or choose “Copy” to copy-and-paste the image into another application.

The images above are licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.