My daughter came home with the following problem from her second grade teacher the day after Halloween (reworded slightly):

Count your Halloween candy. How many different ways are there to divide your candy into piles?

As the problem is worded, it is not mathematically clear what is being asked. So, I am going to word this questions in two ways:

If you have N pieces of identical candy, how many ways are there to group this candy into M groups? In other words, how many ways are there to divide N things identical things amoung M people? For example, how many ways are there to divide 10 pieces of candy amoung 3 different people?

Now assume that the candy is all different. Now how many ways are there to divide N pieces of candy into M groups?

(The general answer for both of these questions is clearly beyond a second grader, but a second grader can surely handle some specific cases.)

Take a "45-45-90" triangle and subdivide it into smaller triangles such that each smaller triangle is an acute triangle (each angle is acute).

What is the minimal number of smaller triangles needed for this subdivision?

triangle-missing_hole-prob.gif
Explain the following image:

arctan_puzzle-prob.gif

Explain the following image (thanks to Paul Zeitz):

primecard-prob.jpg

An algebra student is just learning to work with fractions and discovers that (s)he can just cancel numbers. For example, canceling the 6 is correct: 
16/64=1/4
Similarly, 
139/695=13/65
Simplifying fractions couldn't be easier.

Find all two and three digit fractions that this "method" of canceling gives correct results.

A polydivisible number is such that its first N digits are evenly divisible by N. E.g., if the number has 5 digits, it must end in 5 or 0. Three-digit numbers must have digital root of 3, etc.

528456 is polydivisible:
52/2 == 0 mod 2, 528 == 0 mod 3, 5284 == 0 mod 4, 52845 == 0 mod 5, and 528456 == 0 mod 6.

  1. What is the least whole number N such that 10,000,000 times N is a polydivisible number?
  2. Is there a finite number of polydivisible numbers?
  3. Has the largest polydivible number, if such exists, been found?
  4. Can there be a 10-digit polydivisible number with all even digits?
  5. Can any polydivisible # exist with the same digit repeated at least four consecutive times? (E.g., 4444)
  6. Can a polydivisible number have more than 10 digits? More than 20 digits?
  7. Given any polydivisible number with less than 10 digits, is it always possible to 'tack on' another digit, thus forming a new polydivisible number?
Thanks to John Morse of Delmar NY

Math Circle
Washington University