Variations on Basic Themes: Digit Problems in the Key of A or G Minor?

Update: Answers, solutions have now been posted in the comments.

I thought of these variations on the well-known combinatorial problems involving 3-digit numbers that pop up frequently as I was teaching arithmetic and geometric series yesterday.
These questions are appropriate for grades 6-12 provided students are given definitions and some practice with arithmetic and geometric sequences, topics that are well within the abilities of middle schoolers. A quick intro to these sequences is all that is really needed OR, as I did below, they can be defined in the problem itself. Thus, these questions provide both practice in arithmetic skills and in combinatorial thinking. Of course, all the experienced or budding programmers out there can write simple code to have their graphing calculators count these, but that should only complement and verify their results, not replace the reasoning needed to solve them, unless these are used for a computer science class (even then, programmers should independently verify their code by solving the problems!).
These are not highly challenging and therefore can be used as Problems of the Day, for extra credit, or enrichment. Our readers will hopefully suggest other extensions and further variations (some are suggested below).

1. The digits of 246 form an arithmetic sequence from left to right because 4-2= 6-4. How many positive 3-digit integers satisfy this condition?


2. The digits of 248 form a geometric sequence from left to right because 4/2 = 8/4. How many positive 3-digit integers satisfy this condition?

Now, how could we make these more challenging? 4-digit numbers or will that make one or both easier, i.e., fewer possibilities? What if the digits were allowed to form these sequences in any order? BTW, I apologize for the music pun in the title. I hope you will respond to that with a positive tone!

Developing Algebraic Reasoning

The following sequence of problems deals with a fairly well-known pattern. Similar questions have appeared on SATs, on other standardized tests and in texts. The intent here is to provide an extended activity for students of diverse math backgrounds and abilities to develop a systematic approach to analyzing patterns. Students should also be encouraged to make a table of values in which the first column is the number of 'crosses' and remaining columns are reserved for other 'dependent' variables. This function-based approach is also an essential feature of this development.

Notes: There are many ways to approach these questions. Encourage students to share theirs! These questions involve pattern-based thinking, combinatorics, recursive sequences, arithmetic sequences and algebraic reasoning. Parts (d) and (e) are more challenging for some. Based on the pattern of the first 3 or 4 terms, some students will simply develop a linear formula of the form aN+b for the perimeter (which is somewhat harder than the area). It is important for our prealgebra and algebra students to recognize that any arithmetic sequence like 12,20,28,36,… can be described this way. My experience is that if a class has 20 students, there will be at least 5 different ‘counting’ methods discussed, Students often are very creative here and not all use ‘linear’ thinking!

Problems 2-1-07

PLS NOTE: BLOGGER HAS APPARENTLY BEEN DOWN FOR MOST OF THE DAY. I APOLOGIZE FOR ANY INCONVENIENCE THIS MAY HAVE CAUSED. I WILL TRY TO PUT UP NEW PROBLEMS FOR 2-2-07 BUT NO GUARANTEES AS TO IF OR WHEN...



Just 3! Have fun!!

Note for #1: Assume k is a positive integer.

Problems 1-31-07

Tomorrow's problems focus on sequences and are of varying levels of difficulty. Although #4 may be more appropriate for Algebra 1/2 students, middle schoolers should be able to handle the others. Again, read the comments later in the evening for the answers, comments and solutions. There were some profound ideas expressed about today's questions particularly that innocent-looking quadrilateral problem with the 60 degree angles!