Last week’s Tolstoyan problem is a good example of how important it is, to simplify calculations, to have a proper approach (i.e., ad hoc). If we were to call, for example, x the surface area of the larger field and y the number of harvesters, we would obtain a rather cumbersome system of equations, which Reyes Luelmo simplifies considerably with his ingenious statement: “1 harvester x 1 day is the unit; the number of harvesters, n. The small field needs n/2 x 1/2 day + 1. The large one, nx 1/2 day + n/2 x 1/2 day more. And since the large one is twice the size of the small one,
2(n/4 +1) = n/2 + n/4, where n = 8″.
And since we are going to dust off problems linked to great thinkers, let’s look at a couple that sparked the interest of two of the greatest physicists of all time: Isaac Newton and Albert Einstein.
Newton’s Meadows
From Tolstoy’s fields to Newton’s meadows.
In fact, this problem attributed to Newton is of popular origin, but in its day it caught the attention of the great scientist, who wrote about it (and devised a variant that we will see later), and since then it has borne his name:
Three meadows covered with grass of the same thickness and growth rate have the following areas: 3 1/3 hectares, 10 hectares and 24 hectares. The grass in the first meadow is eaten by 12 oxen for 4 weeks, and in the second by 21 oxen for 9 weeks. How many oxen will eat the grass in the third meadow for 18 weeks?
The problems of pastures and cattle are as abundant as those of shepherds and sheep, and surely go back to the origins of livestock farming; but this one has the peculiarity that the grass continues to grow while they eat it. That is why it caught the attention of Newton, who in his Arithmetic Universalis proposes the following variant:
Knowing that 75 oxen have eaten the grass of a meadow of 60 ares in 12 days and that 81 oxen have eaten the grass of a meadow of 72 ares in 15 days, how many oxen are needed to eat the grass of a meadow of 96 ares in 18 days? It is assumed that the grass in all three meadows was of the same height and that the grass continues to grow uniformly.
And if you haven’t had enough of Newton’s oxen, you can entertain yourself with Archimedes’ countless bulls by revisiting an installment from last year: The Flock of the Sun (3 3 2023).
The clock with deformable hands
In relation to someone who demonstrated that time stretches and contracts, nothing is more appropriate than a problem in which this happens to the hands of the clock.
One day when Einstein was lying sick in bed, his friend and biographer A. Moshkovskii proposed the following problem to distract him:
Let us consider a clock that is marked at 12 o’clock. If in this position the hour hand and the minute hand were to exchange their functions (i.e. if the hour hand were to lengthen and the minute hand were to contract), the time would be the same; but at other times, for example at 6 o’clock, this exchange would give rise to an absurd situation, which could never occur in a normally functioning clock: the minute hand could not be at 6 when the hour hand is at 12, as in Dali’s famous “Soft Clock”. But there are other times, besides 12 o’clock, when the clock hands exchanging their lengths would produce situations that can occur in a normally functioning clock (although only if the hands were superimposed would it still be the same time). How many and what are these moments of interchangeability of the lengths of the clock hands?
According to Moshkovskii, Einstein did not need more time to solve the problem than it took him to formulate it. So you’re already late…
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