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General Science => General Science => Topic started by: alancalverd on 14/12/2022 23:21:07

Title: Advent calendar maths question!
Post by: alancalverd on 14/12/2022 23:21:07
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? 
Title: Re: Advent calendar maths question!
Post by: Eternal Student on 15/12/2022 00:17:36
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.
Title: Re: Advent calendar maths question!
Post by: Eternal Student on 15/12/2022 00:26:48
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.
Title: Re: Advent calendar maths question!
Post by: vhfpmr on 15/12/2022 00:37:42
0,1,2,3,4,5

0,1,2,6/9,7,8
Title: Re: Advent calendar maths question!
Post by: alancalverd on 15/12/2022 10:23:29
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.

Title: Re: Advent calendar maths question!
Post by: Bored chemist on 15/12/2022 12:14:17
Interesting. It's a bit like Sudoku. the "numbers" are not really numbers.
Title: Re: Advent calendar maths question!
Post by: Eternal Student on 16/12/2022 22:57:21
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.
Title: Re: Advent calendar maths question!
Post by: Petrochemicals on 17/12/2022 06:02:11
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!
Title: Re: Advent calendar maths question!
Post by: alancalverd on 17/12/2022 09:33:29
Now that is true genius!
Title: Re: Advent calendar maths question!
Post by: paul cotter on 17/12/2022 09:43:17
Wow!, Alancalverd, the notorious atheist and iconoclast discussing advent. What a turnaround.
Title: Re: Advent calendar maths question!
Post by: alancalverd on 17/12/2022 10:12:51
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.
Title: Re: Advent calendar maths question!
Post by: Bored chemist on 17/12/2022 13:19:28
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
Title: Re: Advent calendar maths question!
Post by: alancalverd on 17/12/2022 14:03:15
Athenian aircraft carriers?

Quote
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.
Title: Re: Advent calendar maths question!
Post by: Petrochemicals on 17/12/2022 20:22:41
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?
Title: Re: Advent calendar maths question!
Post by: chiralSPO on 17/12/2022 22:08:29
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
Title: Re: Advent calendar maths question!
Post by: Bored chemist on 18/12/2022 10:44:35
Athenian aircraft carriers?
Best sort (at the time).
I'm assuming someone took a kite + flew it from a ship.
Title: Re: Advent calendar maths question!
Post by: alancalverd on 18/12/2022 13:41:40
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.
Title: Re: Advent calendar maths question!
Post by: Petrochemicals on 18/12/2022 15:41:41
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.
Title: Re: Advent calendar maths question!
Post by: chiralSPO on 18/12/2022 16:06:06
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
Title: Re: Advent calendar maths question!
Post by: alancalverd on 18/12/2022 16:18:37
You cheated by counting thumbs as fingers!
Title: Re: Advent calendar maths question!
Post by: Bored chemist on 18/12/2022 16:52:43
You cheated by counting thumbs as fingers!
It's hardly "cheating" when everyone does it.
Title: Re: Advent calendar maths question!
Post by: Petrochemicals on 18/12/2022 18:37:58
You cheated by counting thumbs as fingers!
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
3
I could be wrong!
You can actually do away with the "V" on the 2a) cube, as long as you substitute it for a blank face. If you are willing to hide the one,

1a)x,v, xx, xv, ix
2a) I, ii, iii,iv, x
Title: Re: Advent calendar maths question!
Post by: alancalverd on 20/12/2022 11:21:42
Oh no you can't.

Exercise works pretty well, though I'm not allowed to advertise the company I work with either!