[alancalverd]

This might amuse those of a mathematical persuasion.

We have a novelty advent calendar. For anyone brought up in a different faith, it counts down the days from 1 to 25 December. Ours consists of two numbered wooden cubes which you can rearrange to display every integer from 25  to zero.

What numbers are painted on each cube? 

[Eternal Student]

Hi.

   I don't think it can be done.   Specifically the single digit numbers aren't always displayed  as  09,   08,  ..., 00  (i.e. strictly using both cubes)  some of them are just being displayed with one cube and the other one being put away in a cupboard somewhere.   That is a little deceptive.

   Accepting that possibility the following is one suitable arrangement of numbers of the cubes:

Spoiler: show

   Cube 1:      0,  1,  2,  3,  4,  8
    Cube 2:     1,  2,  5,  6,  7,  9 

There are probably other arrangements that would also work.

Thank you for the entertainment and Merry Christmas.

[Eternal Student]

Hi again.

Hmm...  maybe someone would come up with a solution where the number  6   can be turned and used as the number 9.   I'll leave that possibility for someone else.

Best Wishes.

[vhfpmr]

0,1,2,3,4,5

0,1,2,6/9,7,8

[alancalverd]

And now, a couple of riders for amusement:

1. Show your working! Rigorous formal proof, please, not simple enumeration!

2. If there were no requirement to display 00, could you use a "proper nine"?

I'm reminded of the (possibly apocryphal) history of integer programming. Knowing, in depressing detail, the combat survival and effectiveness of various ships, US Navy mathematicians were tasked to calculate the optimum fleet requirement and disposition to invade Japan. The problem with the analytic solution is that 0.83 of an aircraft carrier, for instance, won't float, and if you add or subtract one you have to rethink all its escorts and tankers...... So they had to invent an entirely new branch of mathematics.

[Bored chemist]

Interesting. It's a bit like Sudoku. the "numbers" are not really numbers.

[Eternal Student]

Hi.

And now, a couple of riders for amusement:

1. Show your working! Rigorous formal proof, please, not simple enumeration!....

    Are you sure?   It's probably not going to be interesting reading.  The puzzle is actually quite nice because once you exhibit a solution, it's quite easy to check that this is actually a valid solution and you don't have to bore everyone.   Compare with what computer scientists describe as an "NP problem".
    Moreover, you can't "un-see" the solution and completely avoid using it to produce a sleek proof that wasn't really anything like the actual technique you used at first to get the solution.  The educational value is then missing, since the advent cube calendar result isn't really an important result in mathematics in its own right.   Meanwhile, if you really did present the full reasoning that was first applied, everyone is bored and goes to watch a film on the TV.

Here's the reasoning I applied and I can't be bothered to polish it into a more formal and sleek proof  (meanwhile there is a film you could watch instead):
   Some dates give a lot of information.   Consider those with repeating digits   22,  11  and 00.
Those have already told you that both cubes must have the digits  0,1 and 2.
A cube only has 6 faces.   So there are only  2x3 = 6 more digits you can have.
Just by looking at the last digit in the dates  10 through to 19  you know you need the digits 0 to 9 (which is 10 digits).
You need 10,  you can have only 9   ( 0,1, 2 plus 6 more).    That's it.  The puzzle can't be solved.

Did anyone do it differently?

To move beyond that you know there must be a trick,  e.g.  using only one cube for single digit dates or turning a 6 upside down etc.

Best Wishes.

[Petrochemicals]

Because you need a double zero it is impossible.

This might amuse those of a mathematical persuasion.

We have a novelty advent calendar. For anyone brought up in a different faith, it counts down the days from 1 to 25 December. Ours consists of two numbered wooden cubes which you can rearrange to display every integer from 25  to zero.

What numbers are painted on each cube? 

I believe it is possible with roman numerals.

1a) no face, X, V, IX, XX, XV
2a)I,II, III, IV, V, X

I could be wrong!

[alancalverd]

Now that is true genius!

[paul cotter]

Wow!, Alancalverd, the notorious atheist and iconoclast discussing advent. What a turnaround.

[alancalverd]

Occasionally, one observes without comment.

I'm embarrassed to be a member of this species every time I come across an unnecessary countdown, but the occasional flash of inspiration like Petro's Roman numerals restores some hope that all is not lost.

[Bored chemist]

If you use base six the problem is trivial; 2 dice numbered 0 to 5 will do.
If you want to lose the leading zero (i.e. have the last day shown as "1" rather than "01") you can replace the 6 on one of the dice with a blank face (and / or the 5 if you like).

The problem with the analytic solution is that 0.83 of an aircraft carrier, for instance, won't float, and if you add or subtract one you have to rethink all its escorts and tankers...... So they had to invent an entirely new branch of mathematics.

As is often the case, the Greeks got their first.
https://en.wikipedia.org/wiki/Diophantine_equation

[alancalverd]

Athenian aircraft carriers?

While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (beyond the case of linear and quadratic equations) was an achievement of the twentieth century.

[Petrochemicals]

Occasionally, one observes without comment.

I'm embarrassed to be a member of this species every time I come across an unnecessary countdown, but the occasional flash of inspiration like Petro's Roman numerals restores some hope that all is not lost.

Golly gosh Alan, I've gone all red.

Is yours in roman numerals?

[chiralSPO]

If you use base six the problem is trivial; 2 dice numbered 0 to 5 will do.

Indeed.

At one time I won a (non-monetary) bet that I could count over 1000 on my fingers. The pour soul didn't say I couldn't use base 2. :-D

[Bored chemist]

Athenian aircraft carriers?

Best sort (at the time).
I'm assuming someone took a kite + flew it from a ship.

[alancalverd]

kitehistory.com/Miscellaneous/Warkites.htm  suggests that military use of kites did not appear in the West until much later, but history is always open to revision.

[Petrochemicals]

If you use base six the problem is trivial; 2 dice numbered 0 to 5 will do.

Indeed.

At one time I won a (non-monetary) bet that I could count over 1000 on my fingers. The pour soul didn't say I couldn't use base 2. :-D

We can only get to 256 where I live.

[chiralSPO]

If you use base six the problem is trivial; 2 dice numbered 0 to 5 will do.

Indeed.

At one time I won a (non-monetary) bet that I could count over 1000 on my fingers. The pour soul didn't say I couldn't use base 2. :-D

We can only get to 256 where I live.

I was using my fingers as a ten-digit binary display. 1111111111 (base 2) = (base 10) 512 + 256 + 128 + 64 + 32 + 16 + 8 +4 + 2 +1 = (base 10)  = 1023

[alancalverd]

You cheated by counting thumbs as fingers!