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The Merchant's Puzzle

Of the Merchant the poet writes, "Forsooth he was a worthy man withal." 
He was thoughtful, full of schemes, and a good manipulator of figures. 

"His reasons spake he eke full solemnly. Sounding away the increase of his winning." 

One morning, when they were on the road, the Knight and the Squire, who were riding beside him, reminded the Merchant that he had not yet propounded the puzzle that he owed the company.

He thereupon said, "Be it so? Here then is a riddle in numbers that I will set before this merry company when next we do make a halt. There be thirty of us in all riding over the common this morn. 
Truly we may ride one and one, in what they do call the single file, or two and two, or three and three, or five and five, or six and six, or ten and ten, or fifteen and fifteen, or all thirty in a row. 
In no other way may we ride so that there be no lack of equal numbers in the rows.

Now, a party of pilgrims were able thus to ride in as many as sixty-four different ways. 
Prithee tell me how many there must perforce have been in the company." 
The Merchant clearly required the smallest number of persons that could so ride in the sixty-four ways.




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