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Before it comes to boiling eggs, a note on the logic puzzle from last week.
Several readers have suggested a solution that does without the trick with the playing cards.
I have added this solution in the text - thank you very much for the hints!
If you like your egg soft, you won't cook it that long - for example, only four minutes.
Unfortunately, there are only two hourglasses in the kitchen.
One runs for five minutes, the other for eight minutes.
How do you have to proceed to measure exactly four minutes with these two hourglasses?
We start both hourglasses at the same time.
Whenever an hourglass has run through, we turn it over again immediately.
As soon as the eight-minute clock has run twice, i.e. 16 minutes after the start, the egg is put into the boiling water.
20 minutes after the start of both clocks, the five-minute clock has run through four times - that is the moment to get the egg out of the water again.
It then cooked for exactly four minutes.
The solution can be found by skillful trial and error.
Or we use a method from number theory with which we will find all solutions with certainty.
So that we get exactly four minutes, a multiple of 5 must be 4 larger or smaller than a multiple of 8.
In other words: a multiple of 5 must leave the remainder 4 or -4 when dividing by 8.
Mathematicians use the modulo function to calculate the remainder of a number when dividing it by another number - mod for short.
For example:
21 mod 5 = 1, because when dividing by 5, 21 leaves the remainder 1.
For our hourglass problem, one of the following two equations must be true:
5 * a mod 8 = 4
5 * a mod 8 = -4
Because dividing by 8 only allows eight different remainders (from 0 to 7) and a remainder of -4 is identical to the remainder of +4, we are looking for all solutions to the equation
5 * a mod 8 = 4
For a, on the other hand, we only need to examine values from 0 to 7, because the remainder of 5 * a when dividing by 8 does not change if we increase a by a multiple of 8.
The only possible solution is a = 4.
This corresponds to the solution mentioned at the beginning of 2 * 8 = 16 minutes and 4 * 5 = 20 minutes.
There is a second solution, however, in which you start boiling the egg four minutes later: 4 * 5 = 20 minutes and 3 * 8 = 24 minutes.
There are even an infinite number of other solutions.
Any number a, which is the sum of a multiple of 8 and the number 4, solves the problem.
We can therefore represent a as follows, where b is any natural number:
a = 8 * b + 4
For example, let's take b = 1 and thus a = 12.
This means: after 60 minutes the 5-minute clock has run exactly twelve times.
And after 56 minutes the eight-minute clock ran seven times and after 64 minutes eight times.
Here, too, there are two consecutive periods of exactly four minutes.
I discovered this hourglass problem in the book "Why cows like to graze in a semicircle" by Albrecht Beutelspacher and Marcus Wagner.
If you missed a puzzle from the past few weeks, here are the ten most recent episodes:
Cleverly asked
Enclosed by six circles
x, y unsolved
Floor plan with gaps
Wrong track
Seven in one go
Hundreds of squares
On the brink
Secret access code
Who lies?
Who is telling the truth
How does the sequence of numbers continue?
