The problem is as follows:
In an porcelain jar there is a set of $2016$ cards. These cards have a numbered printed on their faces and those are $1$ to $2016$, assume there is no repetition. These cards have a peculiar characteristic. The cards such that the sum of their numbers are the same are of the same color, and those that have these different sums are of a different color. Using this information find the number of different colored cards there are inside the jar.
The alternatives in my book are as follows:
$\begin{array}{ll} 1.&\textrm{26 cards}\\ 2.&\textrm{27 cards}\\ 3.&\textrm{28 cards}\\ 4.&\textrm{29 cards}\\ 5.&\textrm{25 cards}\\ \end{array}$
I'm a bit confused with this specific question. How exactly should I assess what is it intended?.
From reading at the problem, my interpretation is that there are two kinds of sets of cards.
One group has cards whose numbers add up to the same amount as their partner. And these happen to be the same color
The other group is the opposite. They do not have a pair that adds up to the same number as the numbers that appear to be written on it. And these happen to have a different color
Is this part correct?. If that's so. Then it comes into play the challenging part. How to distinguish which from which set?
What I attempted to diagnose first was the extreme situation:
This happens with $1999$
Since the maximum is $2016$ there will not be $9991$. In other words there is no reverse order. But that was the only card which I identified which must be a different color.
But how about the card below that number:
$1998$, it just happens that $1+9+9+8=27$
and this also occurs with $999$, as $9+9+9=27$. Hence, what to do here?. Does it exist a way to find which group and how many of these cards must have a different color?.
But again, is this the right way to approach this problem?.
Since I feel lost in this question. I'd like someone could help me with the most detailed explanation as possible and be include a step by step analysis of what should be done to solve this problem. Can someone help me?.
The way I understood the problem statement is the way you sketched it; the cards would be coloured according to the sum of their digits. For example, the cards labeled 1, 10, 100 and 1000 would have the same colour and any cards with different sums would have to be coloured differently. For example, you would need a new colour for the cards numbered $2, 11, 101$ etc.
For the case $a = 1, b \in \{ 0,..., 9 \}, c \in \{0, ...,9 \}, d \in \{0, ..., 9\}$ you can get at most $a+b+c+d = 1+27 = 28$. No number abcd with $2016 \geq abcd >2000$ or $abcd < 1000$ can get a higher sum so this means that you need at most $28$ colours to group your cards by the sum of their digits. On the other hand it seems that by choosing b, c , d in the above case with $a=1$ suitably you can sum to any number between 1 and 28, so that there would be in fact cards of $28$ different colours in the jar.