Monday, June 14

# Enigmatic boxes

On the Dyson numbers, which we dealt with last week, here is what our Utopian Messenger has figured out with patience and a computer:

Only these Dysons appear in the top 2 billion whole numbers.

It occurred to me to put them together to see what would happen and … bang!

By concatenating themselves, every Dyson I have encountered spawns another Dyson, with the same multiplier.

This ensures an infinite supply of these numbers, but does not explain anything, it seems to me.

Here is the proof of infinity. The numbers are formatted with a special punctuation to highlight the groups:

102564.102564 x 4 = 4.102564.10256 (parasite)

102564.102564.102564 x 4 = 4.102564.102564.10256

142857.142857 x 5 = 7.142857.14285 (pseudoparasite)

142857.142857.142857 x 5 = 7.142857.142857.14285

230769.230769 x 4 = 9.230769.23076 (pseudoparasite)

You only need to concatenate them once to see that they can be concatenated indefinitely.

(For more details, see comments from last week).

### Boxes, balls and something else

The probabilistic and distributive problems with boxes containing balls of different colors are a classic, and we have seen not a few of them in previous installments; but the boxes, such intriguing and symbolic objects, give for more. For example:

1. A gentleman and his lady, temporarily separated by adverse circumstances, communicate through a messenger. At one point, the lady sends the gentleman a secret message and, so that the indiscreet messenger cannot read it, sends it inside a box closed with a padlock. The gentleman does not have the key and the lady cannot send it to him, as she wears it around her neck with a chain that she cannot break. Neither can the lock nor the box be broken. However, the knight ends up reading the message. How do you get it? (Hint: there is some analogy between the gentleman’s “trick” and a way of exchanging encrypted messages.)

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2. In a rectangular box (orthohedral, to be more precise) there are three 10-centimeter-diameter balls tangent to each other and tangent to the walls, base, and lid of the box. How long are the sides of the box? Each ball is tangent to the other two, and all of them are tangent to at least one of the walls.

3. On one side of a room there are a lot of oranges and on the other there are 10 boxes that we have to fill according to the following rules:

Each box has a different capacity: in the first box there is room for 1 orange, in the second 2, in the third 3 and so on until reaching the tenth box, which holds 10 oranges.

On each trip we can put oranges in all the boxes we want, but putting the same number of oranges in each box.

In each trip we can take from the pile as many oranges as we want; but you have to put them all in boxes, there can be no loose orange.

How many trips will it take, at a minimum, to fill all the boxes?

And you could not miss a typical two-color boxes and balls, like the following:

4. We have three identical boxes, each containing two white balls and one black ball. We randomly take a ball from the first box and put it in the second, and then we take a ball from the second and put it in the third. What is the probability that, after these two operations, when a ball is drawn at random from the third box, it will be white?

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Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn physics’, ‘Damn maths’ or ‘The great game’. He was a screenwriter for ‘La bola de cristal’.