A Geometry Construction Puzzle

Here is an interesting puzzle: given a line segment AB and a line parallel to it, find points on the segment that divide it into six equal parts. However, you are allowed to perform only two actions:

  1. mark a point on a line
  2. Draw an infinite line through two marked points.

We start with two marked points, A and B. Also remember that after a line is drawn by action #2, one can mark points anywhere on that line. The figure below shows a segment AB and a line parallel to it (green). It also shows an arbitrary point C being chosen on the parallel line and then line CA is constructed (blue). Line CA actually extends to infinity in both directions but, to reduce clutter, it shown as a ray.


The puzzle looks difficult because no measurements, compass, or angle measurements are available. However a bit of exploration with an amazing application called Zirkel (C.A.R.), an open-source compass-and-ruler construction program, can leads one to a solution. By the way – “Zirkel und Lineal”in German == “Compass and Ruler” in English.

Of course, proving the construction is correct will take more effort where Zirkel probably will not be of help!

[Note: I know of two different solutions. One is brute force, the other is a bit more elegant.]

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Insightful Thinking

Puzzles can sometimes be solved elegantly with just the right insight. Here are some lateral thinking puzzles to hone your out-of-box thinking.

Answers are hidden between square braces; select the text to make it visible, like: [ this]

  1. A street in downtown goes over a drawbridge spanning a river. As Jane was approaching it, its warning lights started flashing. Its barriers were lowered, blocking the road. Then the drawbridge itself opened. But Jane gunned the motor, held the wheel firmly, aimed at the bridge, and went straight through. Yet there was no mishap. Why not? [Jane was piloting a boat on the river]
  2. The police found a murder victim and they noticed a pair of tire tracks leading to and from the body. They followed the tracks to a nearby farmhouse where two men and a woman were sitting on a porch. There was no car at the farmhouse and none of the three could drive. The police arrested the woman. Why? [The woman couldn’t walk; she was on a wheel chair whose tire marks the police found at the crime site]
  3. Every day Betty travels 40 km in the course of her job. She doesn’t travel in a wheeled vehicle, nor in any kind of surface craft, yet she never has problems with traffic, police, weather, or airports. What kind of job does she have? [Elevator operator]
  4. William spent three days in the hospital. He was neither sick nor injured, but when it was time to leave he had to be carried out. Can you explain? [William was born in the hospital]
  5. William was the least intelligent and laziest boy in the class, yet, when the results were announced, his name was at the top of the list. How did he manage that? [William Abbot was at top of an alphabetically arranged list]
  6. William went around the world in a ship. Yet he was always in sight of land. How? [William was orbiting the Earth in a spaceship]
  7. William was almost done building a house when it suddenly collapsed. He wasn’t injured or upset; he just cleared the area and started to rebuild the house. Is there a natural explanation for his behavior? [He was building a house of cards]
  8. Richard went on a long trip and was gone several weeks. When he returned, he was found floating at sea. What happened? [Richard was an astronaut. When he returned, his capsule came down on the open sea]
  9. A horse walked all day. Two of his legs walked 21 km but the other two only traveled 20km. Why? [The horse walked in a circle at a mill, around the mill stone]
  10. Three people were holding identical blocks of wood. They released the blocks at the same time. The first person’s block fell downwards. The second person’s block rose upwards. The third person’s block stayed where it was, unsupported. Can you explain? [The second person was underwater. The third person was in orbit.]


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A Question of Equilibrium

A frictionless  uniform string is constrained to move along an arbitrary smooth 3D curve in the presence of gravity. You can imagine the string is enclosed in a close-fitting frictionless tube. Given that the two ends of the string are at the same height, show that the string is in equilibrium — that is, it won’t slip off the curve on its own. The equilibrium can be unstable or unstable.


The figure shows a string AB in a vertical plane. The points A and B are at the same height.

This is not a difficult problem; the difficulty lies in figuring out how to even get started when dealing with a curve of an arbitrary shape.

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Find The Saying

Find the saying or aphorism that fits the following descriptions. I could solve some of them; these answers are hidden but can be seen by selecting the text, like this:

Hidden Answer->Like This<- Hidden Answer

0. A man finds a big pile of pyrite crystals in a cave on property he recently bought. He is overjoyed.

->[All that glitters is not gold.]<

1. A man is dancing with a woman at a party. He has already danced the mambo, fox-trot, and a waltz, and he gets ready to do the next one. But as the music starts, his partner informs him that he has to pay her two dollars first for this dance.

->[It takes two to tango]<-

2. The prospector excavated the ore out of the quarry all right, but on coming home promptly forgot  from where it had been mined.

3 Mary chases a bee that got into her house and skillfully catches it as her boyfriend watches in astonishment. He hugs her and says, “you eyes are gorgeous!”

->[Beauty is in the eye of the bee holder]<-

4 A deaf man awakens one day to find that his hearing has returned. The next day he leaves town on vacation to celebrate this miraculous occurrence.

5 A group of lions leave the game before the Autumn hunting season starts.

->[pride goes before the fall]<-

6. A college student is told that his new roommate in the dorm is named Never. Distraught, he walks to the window, opens it, turns and says “it is better this way” and leaps from the 21st floor to his death.

7. Three philosophy professors, Kay Dunsworth, Sara Jones-Hill and Sarah Pinewski get a deterministic (“absence of free will”) thesis named after them.

->[Que Sera Sera]<-

8. Eric used a one cent coupon at the grocery store. At work, next day, he found a one-cent coin by the water cooler.

->[A penny saved is a penny gained]<-

9. Juan’s wife gave him a new awl to use on his job. After a few years of use, the tool was fairly worn out. Juan offered his service for a day to a hardware store owner in exchange for a brand new awl.

->[All for one, one for all]?<-

10 Last batsman Dilip has to just snick the ball and score one run in order for his team to lift the Gold Cup. But alas, there are simply too many fielders there. ->{There’s many a slip between the Cup and Dilip]<-

This is an excerpt from Mukul Sharma’s “Mindsport” newspaper column from many years ago.

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Posted in Puzzle

A Magic Trick with Geometry

I discovered this “magic trick” while painfully solving a related geometry puzzle.

First, some background. Given an arbitrary triangle ABC, draw squares on each side. Label the centers of the squares D, E, F. This forms another triangle DEF. Figure 1 indicates the basic construction (draw a perpendicular bisector half the length of the side ). ABC does not need to be a right angle triangle.


Figure 1: Constructing square centers

Now, it is fascinating to find that if squares are drawn on the second triangle DEF, but the squares are drawn inward rather than outward, the centers land on the mid points of the original triangle ABC. This is shown in Figure 2: JG is the inward perpendicular bisector of side EF, with JG = JE = JF. G is found to bisect side BC.


Figure 2: Outer squares -> Inner Squares -> midpoints

Now, let’s ask what would happen if we started with triangle ABC, but constructed the inner squares instead of outer squares, on it. Thus we have triangle DEF in figure 3 below, constructed on centers of inner squares.


Figure 3: Centers of Inner Squares on Triangle ABC

Now, it is natural to expect that the centers of outer squares drawn on DEF will fall on the midpoints of the original triangle ABC. That is, inner squares -> outer squares -> midpoints. Thus we have figure 4 below, and see that the square centers indeed fall on the midpoints of triangle ABC. outer-inner-squares-fig4

Figure 4: inner squares -> outer squares -> midpoints

So where is the magic trick, the sleight of hand? Well, examine Figure 4 again. Actually, we found the midpoints thus: inner squares -> inner squares -> midpoints.

And that was really amazing to discover, that the constructions break a natural symmetry:

outer squares -> inner squares -> midpoints

inner squares -> inner squares -> midpoints.

Note: the figures are drawn using a beautiful geometry construction software called C.A.R. (Compass and Ruler).

The puzzle

For the reader who wondered what the original puzzle was, it is a set of related puzzles:

(a) Construct outward squares on a triangle ABC of any shape. Let the centers of the squares be DEF. Now, given only DEF, reconstruct the original triangle ABC.

(b) Instead of squares, draw equilateral triangles on sides of ABC. Given the centers of the equilateral triangle, reconstruct the original triangle ABC.

(c) Generalizing further: On side BC with midpoint G, draw a perpendicular GD such that GD = m * GC, where m is a real number. This generates point D. Similarly draw points E and F using sides AC and AB respectively, using the same value of m.Given DEF, reconstruct the original triangle ABC. (setting m  = 1 gives puzzle (a), and so on.)

An alternative approach to draw points D E F is as follows: Draw two lines from the ends of each side at a constant angle to the side. The intersection of these two lines gives the points D E F. Then 45 degree angles gives case (a); 60 degree angles give case (b).


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Drive-in Movie Theater

A drive-in movie theater has a vertical screen whose top and bottom sides are at a height of h1 and h2 meters, respectively. Now if a viewer in a car is too close to the screen, it will look too foreshortened. On the other hand, if the viewer is too far away, the screen gets too small. Perhaps the best “sweet spot” is when the angle subtended by the screen at the viewer’s eyes is largest.

 drive-in-theater diagram

In the diagram, the viewer is at P and looking at the screen AB. Find the distance d such that the angle theta = angle APC is maximum.

This has a quickie solution; no need for calculus

(the finicky may measure the heights h1 and h2 from eye level instead of ground level)

Posted in Puzzle

A Quick Sum

Find the sum of the series involving factorials

1(1!) + 2(2!) + 3(3!) + … + n(n!)

[note: the notation r! means r(r-1)(r-2)…(1), just in case you didn’t know]


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