How many guppies?

From Marilyn Vos Savant’s column in Parade Magazine, June 19, 2016

Say you have 200 fish in an aquarium, and 99 percent of them are guppies. You want to reduce the proportion of guppies to exactly 98 percent. How many should you remove?
— M.S., Winston-Salem, North Carolina

Give it a try.  The answer is surprising.

In this tank of fish, 198 are guppies. If you remove two, they will still constitute almost 99 percent of the fish (196 guppies in a tank of 198 fish equals 98.9898… percent). Even removing 20 guppies won’t help much more. Their proportion will still be 98.8888… percent. You would need to remove 100 guppies to drop their proportion to 98 percent (98 guppies in a tank of 100 fish).

The joy of kite flying

As a youth, I found a special joy in flying kites. Not in just getting one up into the air but in seeing how high and far away they could become. Somehow, there was a special serenity in leaning back against a tree while a breeze was blowing and doing nothing but watching my sentinel in the sky.

Nowadays one can spend a serious amount of money buying fancy kites of various designs. Some are made of plastic and others of cloth. Some have fancy non-aerodynamic shapes but boldly display superheroes or cute little animals.

In my day, kites were purchased from the local dime store. They came in two types: diamond or box. The diamond kit cost ten cents, or maybe it was fifteen, and the box kite considerably more.

The diamond, or Malay, kite consisted of two sticks of balsa wood, one about two thirds as long as the other and bound together by a single loosely fitting metal band. You rotated one of the sticks until they were at right angles to one another, forming a simple cross.

The sticks were packaged rolled up in a very light paper shaped like two triangles placed together base to base. A small strip of each outer edge was folded and pasted over a loop of string that ran around the circumference. The string was visible at each of the four corners of the kite.

Both of the sticks had notches in each end, and the four exposed portions of the string loops were inserted into them. If one pierced a small hole in the paper covering near where the two sticks crossed and tied a kite string around the vertical one, the kite was reputed “ready to fly.”

Well not quite. Although the diamond kite was touted as a tailless kite as originally assembled , in my experience, it was a very poor flyer.

Many of the small plastic kites available today have the same characteristics. If the wind is too light, then it is impossible to get the kite high enough by running to escape the turbulent air near the ground caused by nearby buildings and trees. If the wind is too strong, the kite’s inherent instability causes it careen around and eventually crash breaking the sticks or ripping the plastic or whatever. There is a sweet spot when the wind velocity is just right and the kite will fly, but it is extremely narrow.

From time to time I wonder how many times a parent has taken a young child to a city park planning to introduce them to the joy of seeing their possession defying gravity and soaring into the sky only to give up in frustration after a few dozen failed attempts.

Yes, the wind has to be strong enough that the weight of the kite can become airborne. But more than that, the kite must be stable in all three directions of motion – pitching up and down, yawing from left to right, and rolling from side to side.

If properly rigged and the wind is strong enough, however, a simple diamond kite can be launched by simply holding it front of you with your back to the wind, gently releasing it and letting it rise.


The first thing to do is attach an additional string on the backside of the kite running from one end of the horizontal stick to the other. The length of the string is shorter than the stick itself so that the it becomes bowed in a gentle arc.

Now, when the wind is blowing on the kite, what is called a dihedral angle is produced. A single flat surface is not presented to the wind but two slanted ones. If, because of a swirl of the air, the side of the kite on the left of the vertical stick is pushed away from you, it presents less surface to the wind. At the same time, the rotation causes the right hand side to present more.

Then there is greater wind pressure on the right side than on the left, and the kite rotates back to a position in which the forces are balanced. This is a perfect example of what engineers call “negative feedback” — a system automatically correcting itself when things get out of whack.Dihedral

The more you bow the kite, of course, the more stable it becomes against yawing to the right or left. However, at the same time, less and less surface area faces the wind and the ability of the kite to lift diminishes. A modest amount of bowing is all that is needed for most kites.

Bowing a kite is no secret. Although the diamond kite is reputed to be of Malaysian origin, the Japanese were bowing their kites for centuries as well.


The second thing to do is to disregard any instruction to attach the kite string to the kite at only a single point. If a gust of wind hits the larger triangle of the kite below the point of attachment, there is nothing to stop it from continuing to rise until the entire kite presents very little surface to the wind, and it starts plummeting to the ground. Another gust catches it as it falls and crazy things begin to happen.

Two points of attachment should be used and not just one. Attach a short piece of string to the vertical stick about half way between the horizontal one and the top of the kite. Attach the other end halfway between the horizontal stick and the bottom. Finally attach the kite string itself to this new one producing a triangle when the wind blows.


This bridle, as it is called, also creates a negative feedback. If a gust blows on the bottom half of the kite, the tension in the lower portion of the string becomes greater than the upper and the kite is righted.

Bridle strings too are no secret. The only tricky part is adjusting the point where the kite string is attached to the triangle so that the angle the kite makes in the wind is a good one. Attach too high and the kite flies to horizontally and has little lift. Attach too low and it has too little lift as well.


This one is the killer. Kites sold in stores nowadays have a few little ribbons of plastic streamer hanging at the bottom that serve no useful function at all. The purpose of a tail is to stop the kite from rolling. In my day, the conventional wisdom was that it was the weight of the tail that prevented a kite from starting to pivot from its upright position and start to rotate about the kite string. A length of knotted rags, the heavier the better was the thing to use.

The thing is that if you want a kite to soar so high that you can barely see it, adding a lot of weight is not what should be done. The wind is what holds everything loft, the kite, the tail, and the kite string. The less weight devoted to the kite and tail, the more that can be used to lift string.

I don’t remember if I thought of this myself or one of my parents told me what to do. But here is the secret that will you a champion kite flyer in the eyes of your friends.

Get an old bed sheet, one, say, about eight feet across. Starting at one edge begin tearing a half-inch strip of material away from the sheet. However — this is the important part — do not complete the tear. When you get to about an inch from the other side, stop what you are doing and begin another tear about a half inch separated from the first and going in the opposite direction. Do this about six or seven times and complete the final tear all the way across.Tail


Now you have a light tail that is about 50 feet long, but with no useless knots. It will produce sufficient wind resistance to rotational forces that your kite will be stable against rolls.

The Endorphin Rush

On a blustery day, take a properly built diamond kite to an open area. Lay out the tail in a straight line on the ground. Put your back to the wind and slowly let your kite leave your hand. Watch it majestically rise into the air with no twitching, no bobbing, and no rotation — a stern monarch of the sky. Let all of your string out until it can barely be seen. Get a comfortable place to sit and marvel at what you have done.


In the modern age, there are restrictions against flying kites weighting more than five pounds above 500 feet. Near airports, this applies regardless of weight.

© 2016 Lyndon M. Hardy

The Counter

I finally have finished all the paperwork associated with the shutdown and dissolution of Alodar Systems, Inc. I now can devote full time to writing and blogging. As a start, here is something that resides down my memory lane — long before smart phones and video games.

In my senior year of high school, despite all attempts to put the last visible signs of boyhood aside, occasionally some toy would come out of nowhere that was just too irresistible to ignore.

One day a pal of mine brought to school a little counter that had come off some large machine or another. It was a small cylindrical can about an inch long and a half inch across with a square slit on the side through which one could see portions of six wheels with digits on them, just like the odometer of a car. A short shaft came out of one end, and when you twirled it between your fingers, the unit wheel slowly turned. Ten turns of the unit wheel caused then ten wheel to turn one number and so on.

At recess, one of our little group set the record for number of turns in five minutes, about a hundred or so. For several days, we traded the counter back and forth, slowly pushing the record to new heights, exercising our fingers for endurance and steeling ourselves to ignore the cramps as we twirled the shaft feverishly.

After recess in math class one day, one of us had an inspiration. He took the worn down nearly circular stub of his eraser and crammed it onto the shaft. The teacher was not looking so he quickly stroked the eraser across his desk. When he was done, his eyes widened. The total had increased by ten with a single stroke!

Quickly, he ran the eraser across his desk again and again. A series of brrruuuppps filled the air and the teacher turned from the board to see what caused such a strange sound. My friend easily covered the counter with his hand. Only streaks of eraser dust gave any hint of anything out of the ordinary.

The teacher returned to his lecture, and with more discretion, my friend spent the rest of the hour stroking the device across the desk boosting up the total. At the end of the period, we all clustered around to see what he had done — an increase of almost a thousand for the hour.

Another of us grabbed the counter as we headed for the next class. Getting thirty in five minutes was no longer an interesting challenge. So for the next few days we spent each class hour with one of us stroking the counter across the desk as fast as he dared, again slowly pushing the record higher and higher.

One of us took the counter home for an evening and returned the next day somewhat bleary eyed but proudly displaying a total that was ten thousand higher than when he had received it. The challenge at this new level spurred us all onward, each one taking it home for a day and bringing it back with the change in the number of revolutions greater than ever before.

When it was my turn, my parents were away for the evening. I stroked the eraser across a table for a while but could see that it would be a long evening before I could come close to what the others had done. Looking around the room, I saw my Mother’s sewing machine and in particular the little rubber wheel that one pushed against the bigger chrome band at the end of the machine when threading a bobbin.

I turned on the machine, reved the machine to full throttle, engaged the bobbin winder and pushed the eraser up to contact it. The little counter let out a high-pitched whirl of protest. The unit wheel spun so fast that individual numbers were undetectable. The counter grew hot in my hand, but I gritted my teeth and held on. For some twenty minutes, I kept at it and finally when my fingers could stand the heat no longer I stopped and looked at what I had accomplished.

The next day, with a grin of satisfaction,  I gave the counter to Carl. I had counted over one hundred thousand in a single session! After steering the subject around to how much sewing our mother’s did and establishing that Carl’s mother did not have a machine, I told him what I had done, confident that my record would stand, “It is all a matter of gear ratios,” I said loftily. “You see, the bobbin wheel turns much faster than the big wheel on the machine and the eraser much faster than even that.”

The following day Carl returned with a sad look on his face. Opening his hand he dropped into my palm twisted shards of metal. On some of them, I could see the remnants of wheels with numbers on them. He had taken the physics principle to heart, propped his bicycle up on blocks so that the rear wheel could turn freely and got his little brother to peddle as fast as he could while he put the eraser up to the tire. If a four-inch sewing machine wheel could produce one hundred thousand revolutions in an hour, a twenty-two inch bicycle wheel could do even better.

It certainly could. The centrifugal force was so great that the counter was literally blown apart. The toy was broken and it was time to move on to other things. The old saying that the only difference between a boy and a man is the cost of his toys is not exactly true. They don’t have to be all that expensive.

© 2016 Lyndon M. Hardy