CH. XII] MISCELLANEOUS PROBLEMS 261 



numbers. For the first card serves to divide the numbers into 

 two sets of 64 numbers, numbers on the second card can be 

 chosen so as to divide each of these into two sets of 32 cards, 

 the third divides each of these into two sets of 16 cards, and so 

 on. The numbers must be written on the cards and the windows 

 cut so that after arranging the cards in their proper order, and 

 turning the pack over, the chosen number appears on the back 

 of the seventh card as seen through the windows cut in the first 

 six cards. To arrange the numbers in this way presents no 

 difficulty, but the geometrical problem of cutting the windows 

 is less easy. I give one solution. 



If we work with seven cards one way of preparing them is 

 as follows. We write under the word Yes on the first card, the 

 64 numbers 1 to 32, 65 to 96; on the second card, the numbers 

 1 to 16, 33 to 48, 65 to 80, 97 to 112; on the third card, the 

 numbers in four arithmetical progressions starting from 1, 2, 5, 6, 

 each of 16 terms, with 8 as difference; on the fourth card, the 

 numbers in eight arithmetical progressions starting from 1, 2, 

 3, 4, 5, 6, 7, 8, each of 8 terms, with 16 as difference; on the 

 fifth card, the odd numbers from 1 to 127; on the sixth card, 

 the numbers in four arithmetical progressions starting from 

 1, 2, 3, 4, each of 16 terms, with 8 as difference; and on the 

 seventh card the consecutive numbers from 1 to 64, on this 

 card however the No must be written on the left hand edge of 

 the card and not on its lowest edge. On the back of the last 

 card we must now write the numbers from 1 to 128 in their 

 natural order, 1 to 32 occupying the first quarter, 33 to 64 the 

 fourth quarter, 65 to 96 the second quarter rotated through a 

 right angle, and 97 to 128 the third quarter also rotated through 

 a right angle. The spacing of the figures requires care, but is 

 not difficult if the principle of construction is grasped, and 

 squared paper used. 



The windows should be cut as follows. We will suppose that 

 we use squared paper dividing each card into four equal quadrants 

 with margins. In the first card, we form the window by cutting 

 out the whole of the second quadrant. In the second card, we 

 get two windows by cutting out the top half of the second 



