Our juggler from last week did not make the best decision in crossing such a precarious bridge by throwing his pins into the air, because their reaction to throwing them vertically upwards—as well as their impact when they return to the circus performer’s hands—exerts a force on the bridge greater than their weight at rest. Not even a juggler can break Newton’s third law.
Some readers have suggested using a scale to check this and to quantify the effect. If you have the typical “weighing scales” at home to control the effects of the diet (and a high enough ceiling), you can try—at your own risk—the following experiment: take an object weighing one kilo or more (a carton of any liquid , for example), get on the scale with it in your hands and throw it slightly upward without losing sight of the scale marker; You will notice that at the moment of launching and at the time of recovering the object (if you manage to recover it in flight) the needle moves slightly to the right.
The juggler’s problem is reminiscent of one that was posed many years ago in a physics exam at an engineering school and which, at the time, became famous:
On a scale there is a cage that, when empty, weighs one kilo with a bird weighing 30 grams perched on its rocker. Suddenly the bird starts to flutter around inside the cage. How much does the needle on the scale indicate?
A variation on the same theme:
Now it is a small fish tank with a fish that is on the scale. The fish tank and water weigh one kilo and the fish weighs 30 grams. Suddenly the fish jumps out of the water and falls back into the fish tank. How does this jump affect the scale needle?
And a variation on the variation:
In the fish tank above, with the same amount of water, there is not a fish but an iron ball weighing one kilo resting on the bottom. If you put your hand in the fish tank and take out the iron ball, what does the needle on the scale indicate at the different moments of this action?
The equivalence of the envelope and the bridges
And from a problem about a bridge (in both senses of the preposition) to another about envelopes and parallel bridges:
The famous tour of the 7 bridges of Königsberg It was not possible because the four parts of the city had an odd number of bridges: 5 to one of the islands, 3 to the other, 3 to the right bank of the river and 3 to the left (a total of 14, yes, but is that we have counted each bridge twice). Therefore, no matter where you started, if you crossed all the bridges without passing through any of them again, you had to do something impossible to complete the task: finish the route in the other three parts at the same time (since you visit all of them according to the sequence enter-exit-enter, or enter-exit-enter-exit-enter in the case of the island with 5 bridges). For the Eulerian route to be possible starting in one area and ending in another (as occurs in today’s Kaliningrad), there would have to be two areas with an even number of bridges and two with an odd number.
When moving from Königsberg to Kaliningrad and from 7 bridges to 5, there are 21 different pairs of bridges that could have disappeared (7×6/2). And 15 of these pairs, when they disappear, leave two areas with an even number of bridges and the other two with an odd number. For example, if we eliminate the bridges marked in the figure (which at first glance seem the most dispensable), both islands are left with 3 bridges and both shores with 2. Therefore, we need some additional data to know which of the 15 possible pairs of bridges has been eliminated. What we can say is that, whatever the missing bridges, now the -solvable- problem of the Kaliningrad bridges is equivalent to the well-known one of drawing an open envelope without lifting the pencil from the paper or going over the same line twice. Do you see the equivalence? And don’t say that there are 4 zones in Kaliningrad while the envelope has 5 vertices (why wouldn’t you say that?).
![ENVELOPES, BRIDGES, BIRDS AND FISHES](https://imagenes.elpais.com/resizer/v2/YLAPKKJE35FNHPHG2U5BP6TOSA.jpeg?auth=3626917befb3b310bfbe7a44ca2c28c5b97b27abb93d060996d6a03ae61582a6&width=414)
And since we have been talking about graphs for a couple of weeks, although without barely naming them, I take this opportunity to recommend once again the wonderful and fun book by Clara Grima In search of the lost graphAs I said at the time, when I started reading it I thought: “Why didn’t I write it?”, but when I finished it I said to myself: “It’s better that she wrote it.”
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#Envelopes #bridges #birds #fish