1. A device called "The Mystery Calculator" was found in a Christmas cracker. The calculator consists of six cards, each of which contains 32 numbers between 1 and 63 inclusive. Someone picks a number and then the operator asks them to reveal which of the six cards the number occurs on. The operator adds up the numbers that appear first on each of these cards and the total is the number that was chosen.

It doesn't take much imagination to see that this operates via powers of 2. Card one is 1,3,5,7,9 etc. Card two is 2,3,6,7,10,11 etc, Card three is 4, 5,6,7,12,13,14 etc. If a number when converted to binary has a 1 in position n (counting from the right) then that number appears on card n. This could of course be easily expanded to more cards (eg for numbers between 1 and 255, use eight cards).

What I noticed is that quite a lot of numbers occur on multiple cards but occur in the same position every time they occur:

* 3 is second on cards 1 and 2
* 6 is third on cards 2 and 3
* 7 is fourth on cards 1, 2 and 3
* 12 is fifth on cards 3 and 4
* 14 is seventh on cards 2,3,4
* 15 is eighth on cards 1,2,3,4
* 24 is ninth on cards 4 and 5
* 28 is 13th on cards 3,4 and 5
* 30 is 15th on cards 2,3,4,5
* 31 is 16th on cards 1,2,3,4,5
* 48 is 17th on cards 5 and 6
* 56 is 25th on cards 4,5 and 6
* 60 is 29th on cards 3,4,5,6
* 62 is 31st on cards 2,3,4,5,6
* 63 is 32nd on cards 1,2,3,4,5,6

There are obviously infinitely many of these numbers since 2^m-1 will always be 2^(m-1)th on cards 1,2...(m-1). However they do seem to be rapidly thinning out.

2. After picking up a few that I missed, it looks like they are all the numbers of the form 2^x -2^y, where x-y>=2.

3. In the last few weeks we've had 2-2-2020, 20-2-2020 and 22-2-2020. We'll get the same again in 2022 and then it will be a long wait til 2111 for the next chances to write dates in this form using only two numerals.

2000 and 2002 each also had three such dates, while 1911, 1919, 1991 and 1999 each had twelve.

[Moderation: off-topic reply by antichrist deleted]