### Puzzles for 11.11.11

Some questions we are asking our classes today:

Anand: How many times this millenium will the date consist of a single digit?

Bill: Today, the day, month, and year are the same. In how many days will this happen again?

Angela: How can you get an answer of 0.0909090909… using only one number, but as many times as you want, and basic arithmetic?

What are you all doing?

### Mimi, I am that student.

Yesterday I opened class with this question, taken from the University of Maryland High School Mathematics Competition (2005):

What is the value of    log (2/1) + log(3/2) + log(4/3) + … + log(99/98) + log (100/99) ?

Students who remembered their logarithm properties (from the last time we had class…) saw almost immediately that they could multiply the arguments together and, ta-da, have the numerator cancel with the denominator until it becomes just log (100) = 2.   But there were a few groups who didn’t remember this property.  Well, one group knew they could expand the arguments into log 2 – log 1 + log 3 – log 2… but didn’t follow this logic far enough to solve the problem.  That would’ve been cool, and it’s exactly why I ask students to share ideas that didn’t work– sometimes it’s a bad idea, other times it’s bad execution.

But anyway, I write because one group was kind of sitting and starting at the problem.  I heard one of them say, “You know the first one is going to be the biggest, and the others get smaller… and they are getting closer to 0.”   And I thought: cool!  They just turned this log properties problem into a sequences problem, even looking at what the terms are approaching!   I missed this structure of the terms because I had so quickly calculated the answer.   I had acted exactly like the students Mimi was talking about in her post (A critical mass problem)– I missed subtlety in my haste to get the answer.   It was a little unfortunate that this group couldn’t actually solve the problem with their realization.  It’s always fun when a cool insight actually helps you to solve the problem.  Maybe next time.

### Giuseppe’s Fingers

I assigned the following problem last week in my class:

Giuseppe likes to count on the fingers of his left hand, but in a peculiar way.  He starts by calling the thumb 1, the first finger 2, the middle finger 3, the ring finger 4, and the pinkie 5, and then he reverses direction, so the ring finger is 6, the middle finger is 7, the first finger is 8, the thumb is 9, and then he reverses again so that the first finger is 10, the middle finger is 11, and so on.

One day his parents surprise him by saying that if he can tell them some time that day what finger the number 1,234,567 would be, he can have a new sports car.  Giuseppe can only count so fast, so what should he do?

Here’s how it went down:

Some of my students realized instantly that counting up to 1,234,567 on their fingers wouldn’t be a very effective use of their time (or Giuseppe’s!).  Others counted up to about 180 until they “abandoned ship” on the brute force method. Read More »

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