lo, it is the 6. Or you might have counted, at first, 6 places to the left and turning up the 6th card, have found the 2. Then you would say “If the 2 is in the 6th place counterclockwise, then the 6 is in the 2nd place clockwise.” And counting 2 places from the 0 to the right, you would again find the 6. The same would hold good if Celarent were to call for any other number.
If you want to do this little trick with 13 cards, you must deal them into 5 or 8 piles. You might begin by asking Celarent how many piles she would like the cards dealt into. If she says 2 (or 11), deal them as she commands, and having done so, ask her whether she would now like them dealt into 4 or 9 piles. If she makes you first deal them into 3 or 10 piles, give her her choice afterward between 6 and 7 piles. If she makes you first deal into 4 or 9 piles, give her then a choice between 2 and 11. If she makes you first deal into 6 or 7 piles, give her her choice afterwards between 3 and 10 piles. If she makes you deal them into 5 or 8 piles, lay them down in a circle at once. In doing so let all be face down except the king, which you place face up. The order will be
You ask: “What spade would you like to find?” If she says “The knave,” reply “Then we count to the knave place.” You count and turn up the 3. Then you say “If the 3 is in the place of the Jack counting clockwise, then the Jack is in the place of the 3 counting counter-clockwise.” You can then count round to this place clockwise, and find it is the 10th. So you continue: “And if the Jack is in the place of the 10 counting clockwise, then the 10 is in the place of the Jack counting counter-clockwise.” You count, turn up and find it so. Then you count up to this card clockwise, and go on “And if the 10 is in the place of the 2 counting clockwise, then the 2 is in the place of the 10, counting counter-clockwise.”
The same thing can be done with a full pack of 52 or 53 cards.
We have thus far considered addition and multiplication separately. Now let us study them combined. Take a pack of 11 cards in their proper order. Cut it so as to carry 3 cards from back to face of the pack. That adds 3 to the face-value of the card in any given place.
Now deal them into 5 piles and gather up the piles according to rule. This by itself would multiply the face-value of the card in any given place by 5. But acting after the other operation, if x be the place and y the face-value (or original place) we have
y = 5x + 3.
On the other hand, starting again with the cards in their proper order, if we first deal into 5 piles and then carry 3 cards from back to face, we have
y = 5(x + 3).
In short, the order in which the operations are to be taken in the calculation of the face-values is just the reverse of that of their actual performance. The reason is too obvious to require explanation.
It is easy to see that before dealing the cards out in the little trick I proposed your showing Celarent you can perfectly well allow her to cut the pack first, provided that after the dealing, or at any time, you recut so as to bring the zero card to the face of the pack. This will annul the effect of the cutting.
I want to call your attention, Barbara, to the fact that there is another way of effecting multiplication besides dealing out into piles and gathering in. Suppose for instance you hold in your hand the first 11 spades in their proper order while the first 11 diamonds in their proper order are lying in a pack face down upon the table. We will now effect upon the spades the operation
5x + 3
and simultaneously upon the diamonds the inverse operation
For this purpose begin by bringing 3 spades from the back to the face of the pack. Then bring 5 spades from back to face, lay the face card down on the table face up and in its place put the top diamond. Bring 5 more spades from back of pack to its face, lay the face card down face up upon the other card lying on the table face up, and replace it by the top card in the pile of diamonds. Repeat this process until it can be repeated no more owing to the exhaustion of the pile of diamonds. You will now hold all the diamonds in your hand. Carry 3 cards from the face of the pile to the back, and the whole double operation will be complete.
You can now say “If the 7 of diamonds is the 5th card in the pack of diamonds, then the 5 of spades is the 7th card in the pack of spades,” and, in short, each pack serves as an index to the other.
From the point of view of this proceeding, multiplication appears as a continually repeated addition. Now let us ask what will result from continually repeating multiplication. As before lay the 11 diamonds in their proper order face down on the table, and take the 11 spades in their proper order in your hand. Deal the spades into 2 piles and gather them up. Put the back card, the 2, down on the table and replace it by the top diamond. Again deal the cards in your hand into 2 piles and gather them up, and put the back card (the 4) upon the one lying face up, and replace it by the top diamond. Proceed in this way until you have laid down all your spades except the knave which you never can get rid of in this way. You will now find that the spades run in geometrical progression, each the double of the preceding
2 4 8 5(≡ 16) 10 9(≡ 20) 7(≡ 18) 3(≡ 14) 6 1(≡ 12).
In fact, if, as before, x be the place, y the face-value,
y = 2x.
Then, in the other pack we ought to have
x = log y/log 2.
In fact, for these the face-value is increased by 1 when the place is doubled. For the order is
10 1 8 2 4 9 7 3 6 5.
You may now do this surprising trick. Ask Celarent to cut the pack of diamonds (with the knave of spades but without the knave of diamonds). Then, ask how many piles she would like to have the diamonds dealt into. Suppose, to fix our ideas, she says 5. You obey and gather up the cards according to rule. You then cut so as to bring the knave of spades to the face; and in doing so you notice the face-value of the card carried to the back. In the case supposed it will be 4. Then carry as many cards (i.e. in this case, 4) from the face to the back of the pile of spades. Then ask Celarent what diamond she would like to find. Suppose she says the 3. Count to the 3rd card in the pack of spades. It will be the 6. Then say, “If the 6 of spades is the 3rd card, then the 3 of diamonds is the 6th card,” and so it will be found to be.
I have not given any reason for anything, my Barbara, in this letter. In your family you are very high in reasons and in principles. But if you think I have said anything not true, it will be a nice exercise in the art of reasoning to make sure whether it is true or not.
1. Published at Heidelberg in 1496, at Freiberg in 1503, in Strassburg by Grüninger in 1504, in Strassburg by Schott in 1504, in Basle in 1508; etc.
2
Ribot’s
Psychology of Attention
| 19 June 1890 | The Nation |
The
