A little known mathematician by the name of Charles L. Dodgson invented a game played on a circular billiard table. He is better known as Lewis Carroll, author of Alice's Adventures in Wonderland. The problems for this issue are to find paths of a cue ball as it travels with constant speed on a circular table.  The ball rebounds from the circular cushion such that the angle of incidence is equal to the angle of reflection. The path starts at point A on the cushion with the initial path segment AB at an angle  from the radius OA, where O is the table center and . See Figure 1.  

For some values of , the path returns to point A and then repeats. Each repetition is called a cycle and each cycle has S segments. The rebound points circle the cushion T times during a cycle, For example if  then the cycle is a square with  and . For the cycle shown in Figure 1,  and . Figure 2 shows the first 42 segments of a path with . Geometric sketchpad was used to construct these figures and is a fun program if you have lots of time, email me if interested.

                                Figure 1                                                                              Figure 2                                                                            

Prob. 1    Find  if the cycle is an equilateral triangle.

Prob. 2    Find  for the cycle shown in Figure 1.

Prob. 3    Find S and T for the cycle corresponding to . Part of the path is shown in Figure 2.

Bonus "The Six Pack Problem"     Joe Moser suggested this problem as a warm up, it took me forever to solve it so I have promoted it to the bonus category. A box measures 3x3x3. How can six blocks measuring 2x2x1 be fit into the box? No saws or wood chippers are allowed. Speedy solvers need not report their solution time.

Send your solutions to Herb.Bailey@rose-hulman.edu or to Herb Bailey, Math. Dept., Rose-Hulman, 5500 Wabash Ave., Terre Haute IN 47803.  

The bonus problem in the last issue was to explain the Fido puzzle ( http://digicc.com/fido/ ) and you sent a wide variety of solutions. I have distilled your solutions along with some internet solutions and those of my mathematical friends into the following 'simplest proof'. 

Result 1   If N is a positive integer and S the sum of its digits, then N – S is a multiple of 9.

Let , then . Thus  

. Since  is just k nines (not to be confused with dogs), then  is a multiple of 9. 

Result 2  If N is a positive integer and N* is the number formed by scrambling the digits of N, then the positive difference of N and N* is divisible by 9.

The digit sum S is the same for both N and N*. From Result 1 we have and , where j and j* are integers. Thus the positive difference of N and N* is  or and in either case divisible by 9.   

In the Fido problem the player inputs all but one of the digits of the positive difference of N and N*. The computer sums the digits of the player's input and determines the missing digit so that Result 2 is satisfied.  

Solvers of the problems for previous issue are listed. I should have given credit to Mike Rinker rather than his father Bill for a solution of the tank problem of the summer issue.