Regarding the Dyson numbers, which we dealt with last week, here’s what our Utopian Messenger patiently and PC discovered:

These Dysons only appear in the top 2 billion integer numbers.

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

By putting themselves together, every Dyson they encounter spawns another Dyson, with the same multiple.

This guarantees an unlimited supply of these numbers, but it doesn’t explain anything, it seems to me.

Here is the proof of infinity. Numbers are formatted with special punctuation marks to distinguish 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 assemble them once to see that they can be attached indefinitely.

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

### Chests, balls and something else

Probability and distributive problems with boxes containing balls of different colors are classic problems, and we haven’t seen a few of them in previous installments; But chests, such symbolic and interesting things, give more. For example:

1. A man and his lady, temporarily separated by adverse circumstances, communicate through a messenger. At some point, the lady sends the man a secret message, so that the reckless messenger cannot read it, he sends it inside a box closed with a lock. The master does not have the key and the lady cannot send it to him, because she wears it around her neck with a chain that she cannot break. The lock or box cannot be broken. However, the knight ends up reading the letter. How did you get this thing? (Hint: there is some similarity between the gentleman’s “stunt” and the way encrypted messages are exchanged.)

2. In a rectangular box (autohedral, to be exact) there are three balls of 10 centimeters in diameter interwoven with 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 they are all tangent to at least one of the walls.

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

Each box has a different capacity: in the first square there is space for one orange, in the second square, in the third 3 and so on until reaching the tenth square, where 10 oranges can fit.

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

On each trip we can take from the heap as many oranges as we want; But you need to put them all in boxes, there can be no loose orange.

How many trips are required, at least, to fill all the boxes?

And you can’t miss the 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 box, then we take a ball from the second box and put it in the third box. What is the probability that drawing a random ball from the third box after these two operations will be white?

**Carlo Frappetti ***Writer, mathematician, and member of the New York Academy of Sciences. He has published more than 50 popular scientific works for adults, children, and youth, including “Damn physics”, “Damn maths” and “The great game”. He was the screenwriter of “La bola de cristal”.*

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