The arithmetic of the clock

In the comments of recent weeks, and depending on some of the problems raised recently, the concept of congruence often appears.

In number theory (there is also a geometric congruence), two integers are said to be congruent when they give the same remainder by dividing them by a third, called a module. Thus, 7 and 19 are congruent with respect to 4 because both, when divided by 4, give as remainder 3.

Some congruences are evident; For example, all odd numbers are congruent with respect to 2, since they all give 1 as a remainder when divided by 2 (what can we say, in this sense, of numbers ending in 1?).

More informationJohann Carl Friedrich Gauss, the prince of mathematicians

The congruence relationship is expressed by three parallel strokes and the module in parentheses:

A ≡ B (mod m)

It means that A and B are congruent with respect to M.

Congruence can also be defined as the ratio between two integers whose difference is divisible by a third. If a and b are congruent with m, they give the same remainder, r, when divided by m, from where:

a = pm + r

b = qm + r

where p and q are whole numbers, and therefore:

a – b = (p – q)m

then a – b is divisible by m.

Congruence is the basis of modular arithmetic, introduced by Gauss in the early nineteenth century with his book Disquisitiones Arithmeticae. And modular arithmetic is also known as "clock arithmetic", because clocks illustrate in a very graphic way the equivalence relationship of the hours with respect to modulus 12: thus, 7 and 19 hours are represented in conventional clocks in the same way: with the major hand in 12 and the smallest in 7.

Johann Carl Friedrich Gauss (1777-1855), German mathematician, astronomer and physicist, in a portrait by Christian Albrecht Jensen.

Problematic watches

You can not talk about clock arithmetic without thinking about the numerous problems and riddles (some well known and others not, some easy and others not so much) that have watches as protagonists. They constitute a whole section of the problems of ingenuity, which in turn can be divided into three subsections: needle clocks, hourglasses and digital clocks. Let's look at some of the first type:

A soneria clock, of those that give the hours with chimes, takes 6 seconds to give 6. How long will it take to give 12?

Near my house there are two clocks that tell the hours at different speeds: one gives three chimes in the same time that the other gives two. They are synchronized and start playing at the same time. At what time does the slow clock chime two more when the fast one has stopped ringing? (Based on real events, such as the last one).

At 12 o'clock, the three hands of the clock – the hour, the minute hand and the second hand – coincide exactly (they present weapons to the Sun, as Ramón Gómez de la Serna would say). When will the three coincide again?

And as a culmination, a well-known classic, but of obligatory mention in this context. Classic and historical, because the anecdote is real:

One afternoon, Kant saw that the clock in his house had stopped. Shortly afterwards he walked to visit a friend, at whose house he noticed the time marked by a wall clock. After a good conversation with his friend, Kant returned home by the same way, walking, as usual, with the steady and regular pace that had not changed in twenty years. He had no idea how long it had taken to make the way back, as his friend had recently moved and Kant had not yet timed the journey. However, as soon as he got home he set the clock to time. How did he do it?

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