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[Reasoning Exercises]
| Winter 1887 | Houghton Library |
[Number Series]
The following rows of numbers are called Fermat’s series:
0 1 3 7 15 31 63 127 255 511 1023 2047 etc.
2 3 5 9 17 33 65 129 257 513 1025 2049 etc.
Find out the rules of the succession of numbers in these two series.
The numbers in the following row are called the phyllotactic numbers. The series is also called Fibonacci’s series, because first studied in the XIIth century by the mathematician Leonardo of Pisa, called Fibonacci.
0 1 1 2 3 5 8 13 21 34 55 89 144 etc.
Find out the rule of succession of these numbers.
Do the same for the series
2 1 3 4 7 11 18 29 47 76 123 199 322 etc.
The following series are called Pell’s series:
0 1 2 5 12 29 70 169 408 985 2378 etc.
2 2 6 14 34 82 198 478 1154 2786 6726 etc.
Find the rule of succession for these.
0 1 1 0 −1 −1 0 1 1 0 −1 −1 etc.
Find the rule of succession here.
What is the product of two corresponding numbers in Fermat’s series?
Compare the square of any number in any series with the products of those which precede and follow it. What rules can you find?
If two numbers one over another in any pair of series are both prime, what is true of the number of their place in the series? (N.B. The places are to be numbered beginning with 0.)
What relation can you find between the phyllotactic number whose place in the series is expressed by the sum of any two numbers, say m and n, and the numbers in the mth and nth places in that series and in the series underneath it?
Every third number in the series of Fibonacci is even, every 4th number is divisible by 3. What about every 5th number, every 6th etc.? Is there anything analogous in any of the other series?
Compare the greatest common divisor of two phyllotactic numbers with the greatest common divisor of their places in the series.
Give the rule or plan of each arrangement.
What is the relation between arrangements B and F? What between C and D? What is so related to E?
Show that as B is to A, so is D to B, E backwards to D, C to E backwards, F to C, and A backwards to F.
[Relational Graphs]
Required to make some graphs in which every spot is just like every other and has connections exactly like those of every other spot. See how many different forms of graphs you can make under these conditions, with 3, with 4, and so on up to 12.
Make some graphs in which the spots are of two colors and in each color of two shapes and the lines are of two kinds and every line runs from a spot of one color to a spot of another color and the lines are not barbed. See how many different kinds of graphs you can draw under these conditions beginning with the simplest.
Fig. 1
Fig. 2
Fig. 3
Let the black spots • denote points and the circles ˚ lines, and let the connections signify that points lie on lines. Thus means the point A lies on the line B.
Problem: to draw straight lines so that the state of things represented in Fig. 3 shall be carried out.
Fig. 4
Fig. 5
Fig. 6
Figs. 4 and 6 are substantially the same, but Fig. 5 is different.
Fig. 7. Draw two triangles, having their vertices on three lines that meet in one point. Then the corresponding sides, produced if necessary, meet in 3 points lying on a 10th line. Show how this graph represents the relations either of the points or lines.
Fig. 8. The dots stand for the edges of a cube. Show that the junctions may signify either that two edges meet or that they do not lie in one plane.
Fig. 9
| ⊙ | An instant of time. Joined by barbed line to a later instant. The length of the interval is roughly represented by that of the barbed line. Instants are joined by plain lines to the events that take place about that time. |
| • | A person or firm. Goods. ⊚ A sum of money. |
| A bill. Amount due and goods joined by plain lines. Party making out bill joined by wavy line . Party to whom made out joined by line . | |
| A verbal contract. Negotiators joined by lines . The contract is for the sale of goods joined by plain line, for each payment of sum joined by plain line. Goods to be delivered by party joined by dotted line to party joined by broken line . | |
| Order for goods joined by plain line. Order delivered by party joined by line . Goods to be sent by party joined by dotted line to party joined by broken line . Price to be sum joined by plain line, to be paid as usual at end of month. | |
| Act of delivery by party joined by dotted line of object joined by plain line, to party joined by broken line . | |
| Receipt by party joined by scraggly line, , of bill joined by plain line. | |
| Disappearance of party joined by plain line. | |
| Quarrel between parties joined by plain lines. |
[Card Games]
Take a pack of cards, select from it the spades and the hearts, rejecting only the kings. Arrange each of these suits in sequence,—ace, 2, 3, … 9, 10, Knave, Queen,—the ace being at the back, the queen at the front. Put the hearts on the table in a pile, backs up. Deal off the spades one by one into two piles, turning each card over and laying it down, face up. The cards in this dealing are, of course, alternately placed on the left hand and the right hand piles. But when you come to the last card, which will be the queen, instead of putting it down on the pile where it would regularly go, you put it down on the table, face up, to form the first card of a new pile. In the place where the queen would have gone had you proceeded regularly, now put instead the top card of the pile of hearts, which is the ace, turning it face up. Now cover the pile upon which you have just laid this card, with the other pile of six spades, and take up the combined pile into your hand, faces down. Repeat this operation: that is, deal out the cards you hold in your hands into two piles, until you come to the last card which will be the knave, which you place on the queen, as the second card of that pile, and in place of the knave you put on the second pile, the top card of the pile of hearts, which will be the two. You then cover this pile with the other pile of six, and take up the combined pile, as before. Do this over and over until you have done it twelve times in all, when you will hold all the hearts in your hand, and all the spades will lie in a pile on the table. Now I say that there is a singular relation between the arrangement of the spades and that of the hearts, so that when you have once remarked the secret of it, by examining the spades which you hold in your hand, you can readily tell off the hearts in the order in which they lie on the table. What I ask you to do is, preserving their order, to spread out the spades and the hearts on the table, and try if you can see what this relation between the two orders is.
Take a pack of cards, and arrange them in sequence, proceeding from back to face, as follows:—Spades: ace, 2, 3, … 10, Knave, Queen, King. Diamonds: ace, 2, 3, … 10, Knave, Queen, King. Clubs: ace, 2, … Queen, King. Hearts: ace, 2, … Knave, Queen, King. Thus, the ace of spades will be at the back, and the king of hearts at the face of the pack. Take the pack in your hand, face down, and deal the cards out singly, into five piles, in regular rotation, turning each card face up as you lay it down. Take up the third pile and lay it face up on the first, so as to make one pile of the two. Place this combined pile on the last pile but one, so as to make one pile of them. Take up this still larger pile and lay it on the first pile, so as to combine these. Take up this pile and lay it on the last of the original piles, so as to unite the whole pack. Take up the pack into your hand, faces down, and deal the cards out one by one into six piles in regular rotation, turning them up as you lay them down. Place the fifth pile on the fourth, this united pile on the third, this pile on the second, this on the first, and this again on the last, so as to reunite the whole pack, spread the cards out in four rows, of thirteen in a row, as follows, where the numbers show the places of the cards in the pack before they are spread out.
1 2 3 4 5 6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21 22 23 24 25 26
27 28 29 30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49 50 51 52
You will now find a singular relation between a card and a certain other, in respect to their suits and numbers in the suits, their rows and places in the rows. Try to discover that relation.
Take a pack of 52 cards, and select all the plain cards of three suits. Arrange them in order from 1 to 10, one suit after another, in a pack, beginning at the back with the ace, and a ten at the face of the pack. Then you have thirty cards, and the second suit you can conceive as numbered from 11 to 20, and the third suit from 21 to 30. Now hold the pack in your hand with backs up, and deal them out into a number of piles which may be either 2, 3, 5, 6, 10, or 15. The cards are to be dealt out singly, and each card is to be turned up as it is laid down on the table and you are to deal them out to the different piles in regular rotation. When the cards are all dealt out, you place the first pile on the second, and you place that combined pile on the third, you place that combined pile on the fourth, and so on until you have united the whole pack. Now, I wish to know whether you can find a rule or general statement by which, when you know how many piles the cards have been dealt out into, you can tell what their order will be beginning at the back of the pack, and proceeding to the face. Also, what will be the effect of dealing them out twice or three times.
When you have ascertained the rule asked for in the last problem, take the cards in their original order and deal them out into eight piles; as eight does not divide thirty, it will follow that the last two piles will have each one card fewer than the others. Put the seventh pile on the sixth, put this combined pile on the fifth, put this combined pile on the fourth, put this combined pile on the third, put this combined pile on the second, put this combined pile on the first, and this combined pile on the remaining pile. Now, you will find that the rule which you have already discovered in regard to the order of the cards, holds good in this case. This is because you pick up the piles in this particular order. Now I want to know if you can tell me in what order the piles must be taken up, in order to make this same rule hold good when the cards are dealt out into other numbers of piles than eight.
