In a certain convent there were eight large dormitories on one floor, approached by a spiral staircase in the centre, as shown in our plan.  On an inspection one Monday by the abbess it was found that the south aspect was so much preferred that six times as many nuns slept on the south side as on each of the other three sides.  She objected to this overcrowding, and ordered that it should be reduced.  On Tuesday she found that five times as many slept on the south side as on each of the other sides.  Again she complained.  On Wednesday she found four times as many on the south side, on Thursday three times as many, and on Friday twice as many.  Urging the nuns to further efforts, she was pleased to find on Saturday that an equal number slept on each of the four sides of the house.  What is the smallest number of nuns there could have been, and how might they have arranged themselves on each of the six nights?  No room may ever be unoccupied.

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279.—­THE BARRELS OF BALSAM.

A merchant of Bagdad had ten barrels of precious balsam for sale.  They were numbered, and were arranged in two rows, one on top of the other, as shown in the picture.  The smaller the number on the barrel, the greater was its value.  So that the best quality was numbered “1” and the worst numbered “10,” and all the other numbers of graduating values.  Now, the rule of Ahmed Assan, the merchant, was that he never put a barrel either beneath or to the right of one of less value.  The arrangement shown is, of course, the simplest way of complying with this condition.  But there are many other ways—­such, for example, as this:—­

    1 2 5 7 8
    3 4 6 9 10

Here, again, no barrel has a smaller number than itself on its right or beneath it.  The puzzle is to discover in how many different ways the merchant of Bagdad might have arranged his barrels in the two rows without breaking his rule.  Can you count the number of ways?

280.—­BUILDING THE TETRAHEDRON.

I possess a tetrahedron, or triangular pyramid, formed of six sticks glued together, as shown in the illustration.  Can you count correctly the number of different ways in which these six sticks might have been stuck together so as to form the pyramid?

Some friends worked at it together one evening, each person providing himself with six lucifer matches to aid his thoughts; but it was found that no two results were the same.  You see, if we remove one of the sticks and turn it round the other way, that will be a different pyramid.  If we make two of the sticks change places the result will again be different.  But remember that every pyramid may be made to stand on either of its four sides without being a different one.  How many ways are there altogether?