Fun Stuff

Problems of the Month for February 2012

Solutions for these will be posted next month.

Lots of 7's

What is the last digit of \(77^{77}\)?

Even More 7's

What is the last digit of \(77^{77^{77}}\)?

Last Month's Problems and Solutions

Turning 2012

Imagine an array of red dots extending forever in all directions. A path starts at one dot and moves outward in a spiral manner as shown in the diagram to the right (the starting dot is outlined in blue). After passing through the 2012th dot, how many turns has the path made?

Solution

Turns are made after passing through 1 dot, then after 1 more dot, then after 2 more dots, then after 2 more dots, then after 3 more dots, and so on. To find how many turns we have made after passing through the 2012th dot, we must find when the sum 1+1+2+2+3+3+4+... is less than or equal to 2012, but the next sum in this pattern is greater than 2012. The number of terms in the sum will then tell us how many turns were made.

To find this value notice that a sum such as 1+1+2+2+3+3 can be rearranged to be 1+3+2+2+3+1. This sum can then be regrouped as (1+3)+(2+2)+(3+1). or 4+4+4=3×4.

In a similar fashion 1+1+2+2+3+3+4+4 can be rearranged to be 1+4+2+3+3+1+4+1, and then regrouped as (1+4)+(2+3)+(3+2)+(4+1) or 5+5+5+5=4×5.

In general, 1+1+2+2+3+3+...+N+N=N×(N+1).

Thus, we need to find a number N such that N×(N+1) is close to 2012. We find (by guess and check, for example) that 44×45=1980. This means that 1+1+2+2+3+3+...+44+44=1980. Since this sum has 88 terms in it, from what we noted above, after the 88th turn, the path will have passed through 1980 dots.

Since 1+1+2+2+3+3+...+44+44+45=2025, between the 88th and 89th turn the path will pass through the 2012th dot.

So the path will have made 88 turns after passing through the 2012th dot.

2012 Turns

As you move along the path as shown, number the dots sequentially starting at 1. What is the number of the dot on which the 2012th turn is made?

Solution

From what was explained in the answer to the "Turning 2012" problem, the 2012th turn will be made when the sum 1+1+2+2+3+3+... has 2012 terms. This occurs when the sum is 1+1+2+2+3+3+...+1006+1006. Again from the answer to the "Turning 2012" problem, this number is 1006×1007=1,013,042. Thus, the dot passed hrough when the the 2012th turn is made, is number 1,013,042.

Other Fun Math Web Sites

The Math Forum is a great web resource for students and teachers alike. Ask Dr. Math a question, discuss teaching math with other teachers, follow links to other math web sites, ...

Math Delights is a collection of links to lots of fun math activities for elementary school age students.

Figure This!, Cool Math for Kids, and Fun Mathematics Lessons are all excellent collections of on-line math activities.

MathWorld is a complete encyclopedia of mathematics. If you've ever wanted to know the mathematical definition of a word or phrase, this is the place to go.

The Millennium Mathematics Project is a UK-based program for math education (or as the British would write "programme for maths education"). Among the web sites they produce are

NRICH -- mathematics enrichment resources (puzzles, problems, investigations, games) for elementary through high-school students.

SNAP Math Fair has guidelines and resources for organizing and holding a "Student-centered, Non-competitive, All-inclusive, and Problem-based" Math Fair.

Galileo.org, like the SNAP web site above, has resources and ideas for hosting a Math Fair.

PLUS! -- a mathematics magazine with articles describing the beauty of pure mathematics a well as relating mathematics to the real world.

Mudd Math Facts is a collection of interesting and informative topics for math students in high school and beyond.

Math Puzzle is a great site that provides an ever growing collection of math puzzles of all sorts.

Interactive Mathematics Miscellany and Puzzles has a variety of math topics ranging from elementary to advanced.

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