Saint Patrick Parade Puzzle
During a recent St. Patrick's Day parade an interesting and curious puzzle developed. The Grand Marshal issued the usual notice setting forth that "the members of the Honorable and Ancient Order of Hibernians will parade in the afternoon if it rains in the morning, but will parade in the morning if it rains in the afternoon". This gave rise to the popular impression that rain is to be counted as a sure thing on St. Patrick's Day. Casey boasted that he "had marched for a quarter of a century in every St. Patrick's Day parade since he had become a boy".
I will pass over the curious interpretations which may be made of the above remark, and say that old age and pneumonia having overtaken Casey at last, he had marched on with the immortal procession. When the boys met again to do honor to themselves and St. Patrick on the 17th of March, they found that there was a vacancy in their ranks which it was difficult to fill. In fact, it was such an embarrassing vacancy that it broke up the parade and converted it into a panic-stricken funeral procession.
The lads, according to custom, arranged themselves ten abreast, and did march a block or two in that order with but nine men in the last row where Casey used to walk on account of an impediment in his left foot. The music of the Hibernian band was so completely drowned out by spectators shouting to ask what had become of "the little fellow with the limp", that it was deemed best to reorganize on the basis of nine men to each row, as eleven would not do.
But again Casey was missed, and the procession halted when it was discovered that the last row came out with but eight men. There was a hurried attempt to form with eight men in each row; again with seven, and then with five, four, three, and even two, but it was found that each and every formation always came out with a vacant space for Casey in the last line. The, although it strikes us as a silly superstition, it became whispered through the lines that every time they started off, Casey's "dot and carry one" step could be heard. The boys were so firmly convinced that Casey's ghost was marching that no one was bold enough to bring up the rear.
The Grand Marshal, however, was a quick-witted fellow who speedily laid out that ghost by ordering the men to march in single file; so, if Casey did follow in spirit, he brought up the rear of the longest procession that ever did honor to his patron saint. Assuming that the number of the men in the parade did not exceed 7,000, can you determine just how many men marched in the procession?
The answer is 5039.
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1 solution
Let n be the number of men including the fellow with the limp. Then the smallest this can be would be the least common multiple of 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 , or 2 5 2 0 . So, you'd immediately think, "Okay, 2519 men" as the answer. However, there's that one comment, "...that it was deemed best to reorganize on the basis of nine men to each row, as eleven would not do." What does that mean exactly? Does that mean eleven would not divide into 2 5 1 9 , or that it does divide into it (which it does), but having a formation of men eleven abreast is just not acceptable? If the latter, then the next smallest would be 2 ⋅ 2 5 2 0 − 1 = 5 0 3 9 , and that's the answer. Otherwise, for n < 7 0 0 0 , the solution wouldn't be unique.