I'm sorry for the delay! I was in the Pyrenees and my hostel's wifi didn't work properly :P

Anyway,

HERE are the solutions for the Day 1 ^_^

I very much hope that Olentzero/Apapalpador will bring you everything you deserve (should you don't deserve anything, please, try harder througout 2013 XD )

Best wishes and see you soon!!!!

PS: This is an "ordinary" challenge for these Christmas days :D It is quite interesting!! You'll find out! Deadline: January the 10th.

###
Euler Piruleta vs. Cell
and his Plans for conquering Spain

Poor
Daleks… they have been defeated once more thanks to your invaluable help! Although
Euler Piruleta surely deserves to take some free time, threatens to our beloved
planet won’t stop… What a restless life!!

This
time, **Cell**, who has recently
absorbed those 3ESO-Bilingual-Program-students whose necks have red marks from
previous absorptions by boy/girl/pet-friends, is planning a strike against Spain. Some of his former victims were Angel, Iñigo and Miguel... We all know they were in SUCH pain... XD

To
prevent this attack, Euler needs to regroup all the regions in Spain
in a way such that:

- The regions in each group have no common
borders.

- The least possible number of groups is
used.

Our
hero is quite frightened since he doesn’t seem to know the **least number of groups with the property that the regions in each group
have no common borders.**

Can
you help him?

*Use the map below to get the solution. *__How
many different colours would you need to colour Spain’s map noticing that you can’t
colour two frontier regions with the same colour?__ You **must use the least possible number of different colours!**

**Your solution must include
the least number of colours needed and a coloured map of Spain
supporting your decision.**

__Mathematical background:__

This
story stars in the XIX century…One shinny morning of 1852, Francis Gutrhie asked himself about a
problem he had been thinking of… How many different colours would be needed to
colour a map without two frontier countries having the same colour?

He reckoned what he thought would be the correct answer, but he also
found himself unable to prove it. At this point, he decided to ask one of the
greatest mathematicians of that time: *Augustus
de Morgan*, who also didn’t manage to find the solution.

We had to wait until 1977 when *Appel
*and *Hacken* gave a proof based on
computers simulation.

**Per Gessle's "Spegelboll"**

**A famous tune from Sweden**