If two squares share a common vertex, then the centers of the squares and the midpoints of the linking segments shown form another square.
Discovered by mathematicians Paul Finsler and Hugo Hadwiger.
Podcast Episode 69: Lateral Thinking Puzzles
Here are four new lateral thinking puzzles to test your wits! Solve along with us as we explore some strange situations using only yes-or-no questions.
You Are What You Eat
Invertible Head as Basket of Fruit, c. 1590, by the Italian painter Giuseppe Arcimboldo.
He also personified the elements and the seasons.
Precocious
Robert Patch of Chevy Chase, Md., was only 5 years old in 1962 when he designed a toy truck that could be converted into a flatbed or a dump truck by altering the placement of the axles. His father, a patent attorney, saw that the truck’s design was unique enough to be patentable, so Robert signed the application with an X and had the mark witnessed by a notary public. When the application was granted, Robert became the youngest person in history to receive a U.S. patent.
The publicity meant nothing to him, but it did bring one benefit. He had made his working model from bottle caps, Scotch tape, nails, and old shoe boxes. Someone at US Keds, the shoebox brand, saw the story — and sent him a new pair of sneakers.
An Anatomy Lesson
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Back in 2008 I mentioned that, as a joke, Mark Twain had slipped the word oesophagus into an otherwise innocent short story in 1902. I wrote at the time that he’d said that few people noticed anything amiss, but apparently a few did. In a letter to the Springfield Republican on April 12, Twain wrote, “I will say privately that I expected it to bother some people — in fact, that was the intention, — but the harvest has been larger than I was calculating upon. … It is time for me to speak up and stop the inquiries if I can, for letter-writing is not restful to me, and I am not having so much fun out of this thing as I counted on.”
He quotes two letters. The first is from a public instructor in the Philippines:
My Dear Sir: I have just been reading the first part of your latest story entitled ‘A Double-barrelled Detective Story,’ and am very much delighted with it. In part IV, page 264, Harpers’ magazine for January, occurs this passage: ‘far in the empty sky a solitary “oesophagus” slept, upon motionless wing; everywhere brooded stillness, serenity and the peace of God.’ Now, there is one word I do not understand, namely, ‘oesophagus.’ My only work of reference is the ‘Standard Dictionary,’ but that fails to explain the meaning. If you can spare the time, I would be glad to have the meaning cleared up, as I consider the passage a very touching and beautiful one. It may seem foolish to you, but consider my lack of means away out in the northern part of Luzon.
The second is from a professor at a New England university:
Dear Mr. Clemens: ‘Far in the empty sky a solitary oesophagus slept upon motionless wing.’ It is not often I get a chance to read much periodical literature, but I have just gone through at this belated period, with much gratification and edification, your ‘Double-Barrelled Detective Story.’ But what in hell is an oesophagus? I keep one myself, but it never sleeps in the air or anywhere else. My profession is to deal with words, and oesophagus interested me the moment I lighted upon it. But as a companion of my youth used to say, ‘I’ll be eternally, co-eternally cussed’ if I can make it out. Is it a joke, or am I an ignoramus?
“Between you and me, I was almost ashamed of having fooled that man,” Twain wrote, “but for pride’s sake I was not going to say so. I wrote and told him it was a joke. … And I told him to carefully read the whole paragraph, and he would find not a vestige of sense in any detail of it.
“I have confessed. I am sorry — partially. I will not do so any more — for the present. Don’t ask me any more questions; let the oesophagus have a rest — on his same old motionless wing.”
(From Gary Scharnhorst, Mark Twain on Potholes and Politics, 2014.)
Beginnings
First entry in Victoria’s diary, Aug. 1, 1832, when she was 13:
We left K.P. at 6 minutes past 7 and went through the Lower-field gate to the right. We went on, & turned to the left by the new road to Regent’s Park. The road & scenery is beautiful. 20 minutes to 9. We have just changed horses at Barnet a very pretty little town. 5 minutes past 1/2 past 9. We have just changed horses at St. Albans. The situation is very pretty & there is a beautiful old abbey there. 5 minutes past 10. The country is beautiful here: they have began to cut the corn here; it is so golden & fine that I think they will have a very good harvest, at least here. There are also pretty hills & trees.
Five years later, on the day of her accession, she wrote, “Since it has pleased Providence to place me in this station, I shall do my utmost to fulfil my duty towards my country; I am very young and perhaps in many, though not in all things, inexperienced, but I am sure that very few have more real good will and more real desire to do what is fit and right than I have.”
Van Aubel’s Theorem
Draw any quadrilateral and build a square on each of its sides.
Connect the centers of opposite squares, and these two line segments will be perpendicular and of equal length.
Collage
Glimpses from the first day of the Battle of the Somme, July 1, 1916:
“I was ordered to fetch the breakfast from the kitchens about a mile away. On the way back, in the trench, we came across a covey of young partridges and, as we walked along, we were driving them in front of us. A lot fell in a sump which was full of water so they would surely drown. But I could not see them drown, so I pulled the top off and got them out, put them in my steel helmet and lifted it up to the top of the trench. There, their mother was waiting and she chuckled them all together and off they went, never to be seen by us again.” — Pte F.G. Foskett, 7th Bedfords
“I suppose a shell hole is not the best place from which to admire anything but, believe it or not, waving about just over my head were two full-blown red poppies which stood out in pleasant contrast against the azure blue sky.” — Pte G.E. Waller, Glasgow Boys’ Brigade Battalion
“The worst sights were in our own trenches where some of the badly wounded had managed to crawl. We were not allowed to help any of them, but kept our machine-gun mounted on the parapet in case of a counter-attack. The wounded were trying to patch each other up with their field dressings. A chaplain tore his dog collar off in front of me and, with curses, said, ‘It is a mockery to wear it.'” — Pte C.A. Turner, 97th Brigade Machine Gun Company
“I could see, away to my left and right, long lines of men. Then I heard the ‘patter, patter’ of machine-guns in the distance. By the time I’d gone another ten yards there seemed to be only a few men left around me; by the time I had gone twenty yards, I seemed to be on my own. Then I was hit myself.” — Sgt. J. Galloway, 3rd Tyneside Irish
“Now we came on to a German machine-gun post and there were all the twelve of the crew lying dead around the machine-gun; a short distance away we saw the body of one of our sergeants, formerly one of the king’s footmen who joined up with us at Norwich. He had obviously accounted for the machine-gun crew, before he himself received his death blow. A strange feeling possesses one at such a moment. It seems as if one is detached and merely looking at a scene of carnage from a great distance.” — Pte WC. Bennett, 8th Norfolks
“I then went on to the second-line trench and jumped in, to see a German soldier lying on the parapet. With fixed bayonet I approached, then I saw his putty-coloured face which convinced me he was mortally wounded. The German brought up an arm and actually saluted me. I understood no German language but the poor chap kept muttering two words ‘Wasser, Wasser,’ and ‘Mutter, Mutter.’ It took me a minute or so to realize he wanted a drink of water. The second word I could not cotton on to. I am glad to this day that I gave him a drink from my precious water.” — Pte G.R.S. Mayne 11th Royal Fusiliers
“We are filled with a terrible hate. Our actions are born of a terrible fear, the will to survive. Some of the Germans were getting out of their trenches, their hands up in surrender; others were running back to their reserve trenches. To us they had to be killed. Kill or be killed. You are not normal.” — L/Cpl J.J. Cousins, 7th Bedfords
A company commander in the London Division’s Pioneer battalion was left out of the battle: “My recollection, after all these years, is of being in a trench discussing the rumours, helping with the wounded (we had four men killed) and occasionally lying in a bit of shelter, reading Pickwick Papers and watching the activities of a fat and grey rat.” — Capt. P.H. Jolliffe, 1/5th Cheshires
(From Martin Middlebrook, The First Day on the Somme, 1971.)
Unquote
“When anyone asks me how I can best describe my experience in nearly 40 years at sea, I merely say uneventful.” — Edward Smith, captain of the Titanic, in 1907
Reflections
More maxims of La Rochefoucauld:
- “Those who are deceived by our Cunning don’t appear near so ridiculous to us, as we seem to ourselves, when deceived by the Cunning of others.”
- “The World oftner rewards the Appearance of Merit than Merit itself.”
- “The Calm, or Disquiet, of our Temper depends not so much on Affairs of Moment; as on an agreeable, or disagreeable, Disposition of the Trifles that daily occur.”
- “In Misfortunes we often mistake Dejection for Constancy; and we bear them, without daring to look on them; as Cowards suffer themselves to be kill’d, without Resistance.”
- “How brilliant soever an Action may be, it ought not to pass for Great when it is not the Effect of a Great Design.”
- “There are Crimes which become innocent, and even glorious, thro their Splendor, Number, and Excess: Hence it is, that public theft is call’d Address; and to seize on Provinces unjustly, to make Conquests.”
- “‘Tis easier to appear worthy of the Employments we have not, than of those we have.”
- “‘Tis difficult to love those we don’t esteem; but ’tis no less difficult to love those we esteem much more than ourselves.”
- “Every body speaks well of his Heart, but no body dares speak well of his Head.”
- “Flattery is a sort of bad Money to which our Vanity gives Currency.”
- “Those who are incapable of great Crimes don’t readily suspect others of them.”
- “Fortune breaks us of many Faults, which Reason cannot.”
- “We easily excuse in our Friends the Faults that don’t affect us.”
- “None are so happy, or unhappy, as they imagine.”