Some Geometrical Questions

Excerpts

• Some Geometrical Questions

  I append two or three demonstrations, leading to obviously impossible results, which perhaps may amuse any one to whom they are new. I leave the discovery of the errors to the ingenuity of my readers.

• Some Geometrical Questions

  In fact proofs by superposition should be regarded with considerable distrust unless they are supplemented by mathematical reasoning.

• Some Geometrical Questions

  Rotate the lamina successively through two right angles about the diagonal $OB$ as axis and through two right angles about the side $OA$ as axis, and the required result will be attained.

• Some Geometrical Questions

  Hence the probability that a triangle can be constructed out of the three pieces into which the stick is broken would appear to be $\frac{1}{2}$. This is not true, for actually the probability is $\frac{1}{4}$.

• Some Geometrical Questions

  All places whose heights above the mean sea level are equal are on the same level.

• Some Geometrical Questions

  A proof of the proposition involves difficulties of a high order, which as yet have baffled all attempts to surmount them.

• Some Geometrical Questions

  The mathematical theory for a board of $9$ cells has been worked out completely, and there is no difficulty in extending it to one of $16$ cells: but the analysis is lengthy and not particularly interesting.

• Some Geometrical Questions

  Let $P$ be any point on a cubic. Let the tangent at $P$ cut the curve again in $Q$. Let the tangent at $Q$ cut the curve in $A$.

• Some Geometrical Questions

  Sylvester stated that 9 counters can be placed in 10 rows, each containing three counters; I do not know how he placed them,

• Some Geometrical Questions

  To those who have never looked into the matter it may be surprising that patterns formed by the use of square tiles (of which one-half bounded by a diagonal is white and the other half black) should be subject to mathematical analysis.

• Some Geometrical Questions

  A cube has six faces, and if six colours are chosen we can paint each face with a different colour.

• Some Geometrical Questions

  Three beautiful ladies have for husbands three men, who are as jealous as they are young, handsome, and gallant.

• Some Geometrical Questions

  To obtain a solution we observe that we can cut a sheet of paper so that, when folded properly, it will make a model to scale of the room.

• Some Geometrical Questions

  Suppose the pieces to be arranged originally in circular order, with two contiguous blank spaces, then we always move to the blank space for the time being that pair of coins which occupies the places next but one and next but two to the blank space on one assigned side of it.

• Some Geometrical Questions

  Since there is only one cell on the board which is unoccupied, and since no diagonal moves and no backward moves are permitted, it follows that at each move not more than two pieces of either colour are capable of moving.

• Some Geometrical Questions

  The solution is tolerably obvious. First, move the pieces from $a$ to $A$, from $b$ to $B$, from $c$ to $C$, and from $d$ to $D$.

• Some Geometrical Questions

  Next suppose that the end $A$ is twisted once completely round ( through four right angles) before it is gummed to $B$, then a similar cut produces two interlaced rings.

• Some Geometrical Questions

  If any of my readers think that these results could be predicted off-hand, it may be interesting to them to see if they can predict correctly the effect of again cutting the rings formed in the second and third experiments down their middle lines in a manner similar to that above described.

• Some Geometrical Questions

  Hence to interchange all the pieces will require $15 + (7 \times 15)$ moves, that is, $120$ moves.

• Some Geometrical Questions

  Whoever first gets three (or any other assigned number) of his pieces in three adjacent cells and in a straight line wins.

• Some Geometrical Questions

  Thus at present it is not possible to say what is the maximum number of rows of three which can be formed from $n$ counters placed on a plane.

• Some Geometrical Questions

  If more than two colours are used, the problems become increasingly difficult.

• Some Geometrical Questions

  Take any face of the cube $K$: it has four angles, and at each angle three colours meet.

• Some Geometrical Questions

  The construction and the initial arrangement ensure that at any one time there cannot be more than eight vehicles on the track.

• Some Geometrical Questions

  Let $y$ denote the number of passages from one bank to the other which will be necessary.

• Some Geometrical Questions

  Thus the problem is reduced to finding the way of cutting out the paper which gives the shortest route of the kind.

Equations

• Some Geometrical Questions

  5 \times 13 - 8^2 = 1

  The integer relation on which the paradox of the 64-square board that yields 65 squares depends; similar identities follow from the continued-fraction convergents.

• Some Geometrical Questions

  \frac{1}{2}\pi ab

  The area of the semi-ellipse bounded by the minor axis equals one half of pi times a times b, whatever the dimensions of the curve.

• Some Geometrical Questions

  h &=1 + p_1 + 2p_2 + \dotsb\,

  The number of hills equals one plus the number of single passes plus twice the number of double passes, and so on, by the theorem of Cauchy and Euler.

• Some Geometrical Questions

  d &=1 + f_1 + 2f_2 + \dotsb\,

  The number of dales equals one plus the number of single forks plus twice the number of double forks, and so on.

• Some Geometrical Questions

  w &=2(p_1 + f_1) + 3(p_2 + f_2)) + \dotsb\,

  The number of watercourses equals twice the sum of single passes and single forks, plus three times the sum of double passes and double forks, and so on.

• Some Geometrical Questions

  y = 2n - 1

  For n married couples with a boat carrying x = 4 people (n > 5), the least number of passages y from one bank to the other is 2n - 1 (Delannoy's result).

• Some Geometrical Questions

  x=\pm a

  One of the ten lines whose intersection points, placed as counters, give the 19-counter arrangement in 10 rows of five; the lines are x = ±a, x = ±b, y = ±a, y = ±b, y = ±x.

• Some Geometrical Questions

  y=\pm x

  The diagonal pair of lines through the origin; one of the ten lines used in the 19-counter, 10-row construction.

Problems

No exercises in this chapter.