A Problem for Al

I went into math education later than many, and was disappointed to find that math teachers do not, in general, love to do math problems. So I was fortunate that when I landed in Misawa, Japan to teach, they popped me right next door to an old veteran who loved problems as much as I did, and was very demanding about a clear proof and explanation. Today, after twenty years and thousands of miles between us, Al Harmon is still one of my math correspondents and an educational inspiration. He frequently sends me problems he is working on that he doesn't see a solution, or sometimes just doesn't think he has the most elegant solution. I am always flattered that he would think I could solve a problem that he couldn't,

I have another great internet advisor, Dave Renfro, who is no longer employed as a professor, but who still teaches many of us with his posts and references to old journals. Along the way he took to sharing with me the journals he thought might help me learn a little more about math, education, and their history. A few weeks ago he sent me a stack of reprints from the Philosophical Magazine from around 1825-1830. one insert included the "Examination of the scholars at Wyke-House" , a school in Middlesex, England. The scholars were tested on questions ranging from arithmetic to geometry and trigonometry, and took six days to answer the seventy-four questions. The image at top shows the prize medal presented to the winner. The first question this year asks for the student to "Numerate and "point off in periods, half-periods, etc... the numeral 123456789012345678901234567890 (raise your hand if you know what a half-period is.... and if not, look here).... and the last question asks for a proof that x+1/x = 2 cos(theta) could only have solutions when theta was a multple of 180^o....several asked for proofs of geometric or trigonometric identities.

One caught my eye as the type of problem Al might love, so I wanted to share it here, with a thanks to both Al and Dave for their continuing contribution to my education.

Problem 51, from the 1827 examination: Several detachments of artillery divided a certain number of cannon balls. The first company took 72 and 1/9 of the remainder; The second detachment 144 and 1/9 of the remainder. The third company took 216 and 1/9 of the remainder; the fourth company took 288 + 1/9 of the remainder; and so on; ---- finally it was discovered by the commanding officer commanding the brigade of guns, the the shot had been equally divided. Determine the number of detachments and the number of balls in the pile.

Ok, So the first answer (with explanation) wins a pat on the back and high praise....... I noticed an interesting pattern when solving this problem, so there is a nice generalization to any number of regiments... For instance I could write a similar problem for any given number of regiments.... . Will provide that later, if needed.