My recent blog asked to solve the problem below... if you haven't tried it yet and don't want to spoil the fun, then go to the original post now.... (the problem)
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.
My Solution approach The first person, as I explained will take 72+ (T-72)/9 or 64+T/9 cannonballs. The second will take 144 + [T-144 - (64-T/9)]/9 . This second expression simplifies somewhat to 1088/9 +8T/81. If the shares are to be equal, these must be the same quantity. The solution is pretty direct as shown here.
Once we know the total number of cannonballs, we just calculate the number that the first division takes in its turn, and then we can determine how many such shares there will be to distribute. The Total will expire after eight batteries have removed their share, each totaling 576 cannonballs, if the 72, 144, 216, pattern continues..Here is the way it looked on a spread sheet