More Geometry Trouble

http://books.google.com/books?id=kI1aAAAAMAAJ&printsec=frontcover&dq=subject:%22puzzles%22&lr=&num=20&as_brr=1#PPA264,M1

Lewis Carroll offered this proof that all triangles are isosceles:

Let ABC be any triangle. Bisect BC at D, and from D draw DE at right angles to BC. Bisect the angle BAC.

(1) If the bisector does not meet DE, they are parallel. Therefore the bisector is at right angles to BC. Therefore AB = AC, i.e., ABC is isosceles.

(2) If the bisector meets DE, let them meet at F. Join FB, FC, and from F draw FG, FH, at right angles to AC, AB.

Then the triangles AFG, AFH are equal, because they have the side AF in common, and the angles FAG, AGF equal to the angles FAH, AHF. Therefore AH = AG, and FH = FG.

Again, the triangles BDF, CDF are equal, because BD = DC, DF is common, and the angles at D are equal. Therefore FB = FC.

Again, the triangles FHB, FGC are right-angled. Therefore the square on FB = the [sum of the] squares on FH, HB; and the square on FC = the [sum of the] squares on FG, GC. But FB = FC, and FH = FG. Therefore the square on HB = the square on GC. Therefore HB = GC. Also, AH has been proved equal to AG. Therefore AB = AC; i.e., ABC is isosceles.

Therefore the triangle ABC is always isosceles. Q.E.D.

| Science & Math

Out With a Bang

http://en.wikipedia.org/wiki/File:James_Otis.jpg

Lawyer James Otis was a hero in American politics before the revolution. In his later years he used to tell his sister, “I hope, when God Almighty in his righteous providence shall take me out of time into eternity, that it will be by a flash of lightning.”

On May 23, 1783, he was standing in a doorway during a thundershower, telling a story to his family, when his wish came true.

“No mark of any kind could be found on Otis,” ran one account of his death, “nor was there the slightest change or convulsion on his features.”

“This flash of lightning was the first that came from the cloud, and was not followed by any others that were remarkable.”

| Death

Cast Away

Here’s a paragraph from Robinson Crusoe. It contains a remarkable error — can you spot it?

A little after noon, I found the sea very calm, and the tide ebbed so far out, that I could come within a quarter of a mile of the ship; and here I found a fresh renewing of my grief: for I saw evidently, that if we had kept on board, we had been all safe–that is to say, we had all got safe on shore, and I had not been so miserable as to be left entirely destitute of all comfort and company, as I now was. This forced tears from my eyes again; but as there was little relief in that, I resolved, if possible, to get to the ship–so I pulled off my clothes, for the weather was hot to extremity, and took the water. But when I came to the ship, my difficulty was still greater to know how to get on board; for, as she lay aground and high out of the water, there was nothing within my reach to lay hold of. I swam round her twice, and the second time I spied a small piece of rope, which I wondered I did not see at first, hang down by the fore-chains, so low as that with great difficulty I got hold of it, and, by the help of that rope, got up into the forecastle of the ship. Here I found that the ship was bulged, and had a great deal of water in her hold, but that she lay so on the side of a bank of hard sand, or rather earth, and her stern lay lifted up upon the bank, and her head low almost to the water: by this means all her quarter was free, and all that was in that part was dry; for you may be sure my first work was to search and to see what was spoiled, and what was free, and first I found that all the ship’s provisions were dry and untouched by the water: and being very well disposed to eat, I went to the bread-room and filled my pockets with biscuit, and ate it as I went about other things, for I had no time to lose. I also found some rum in the great cabin, of which I took a large dram, and which I had indeed need enough of to spirit me for what was before me. Now I wanted nothing but a boat, to furnish myself with many things which I foresaw would be very necessary to me.

(He does explain later that “I found the tide begin to flow, though very calm; and I had the mortification to see my coat, shirt, and waistcoat, which I had left on the shore, upon the sand, swim away. As for my breeches, which were only linen, and open-kneed, I swam on board in them and my stockings.”)

| Literature · Puzzles

Bacon Testimony

http://books.google.com/books?id=q24oAAAAYAAJ&pg=PA128&dq=1451+lausanne+leeches&as_brr=1&ei=jDxDSbeGHoHwMu3HlOgN

Among trials of individual animals for special acts of turpitude, one of the most amusing was that of a sow and her six young ones, at Lavegny, in 1457, on a charge of their having murdered and partly eaten a child. … The sow was found guilty and condemned to death; but the pigs were acquitted on account of their youth, the bad example of their mother, and the absence of direct proof as to their having been concerned in the eating of the child.

— Robert Chambers, The Book of Days, 1864

| Oddities · Society

The Clairvoyant Penny

http://commons.wikimedia.org/wiki/File:United_States_penny,_obverse,_2002.jpg

Mathematician Thomas Storer offers a foolproof way to foretell the future: Flip a penny and ask it a yes-or-no question. Heads means yes, tails means no.

How can you be sure the answer is accurate? Simple: Flip it again and ask, “Will your present answer have the same truth value as your previous answer?”

  • Suppose the answer is yes. This is either true or false. If it’s true, then the original response was true. If it’s false, then the truth value of the original response is not false, i.e., it’s true.
  • If the answer to the second question is no, this too is either true or false. If it’s true, then the original response was true. If it’s false, then the original response was not false, i.e., true.

Since all the outcomes agree, the penny’s original response is guaranteed to be correct.

| Science & Math