How did my younger sister survive?

Question

In a far other universe, my little sister was trapped in a cave. Surprisingly for humans, in this universe, all caves are electronic. This electronic cave had nothing but 5 buttons, two little screens, one red and one blue, a numeric keypad, and the most brought to light, this little message:

Message

Press the wrong button and that'll be your last action, press the right button and leave this cave.

Screens

The red little screen output the number 11. The blue output 19-8-7-10.

Buttons

The five buttons had 5 different colors. My sister had always been an expert for colors (as well as having a PhD in colors (what a strange universe...)) and she could perfectly remember these:

1. Buff
2. Gold
3. Corn
4. Opal
5. Puce

What my sister did

My favourite

To escape, my sister first pressed her favourite number into the numeric keypad: 2. Immediately, the blue screen started blinking. And 5 seconds after, it output the letter y. Simultaneously, the red little screen changed and output 10. At that moment, she understood she had to write down on a little piece of paper she had: 11 and 19-8-7-10.

That's not here for nothing

She then tried 19, 8, 7, 10 but those four numbers didn't change the blue output and more stressful than anything, after that the red screen output 6...

I was born the...

She tried 17 march 03 and thankfully the blue showed:

17: v
3: u

The red showed 4...

Our cat was born the...

She tapped 25 november 11...

25: s
11: p

Red..... 2...

Father was born the...

She finally tried 29 november 77

29: q
77: m

How did she survive?

---

Alright, that's been four days that this puzzle is still unbreakable :p. Here is the first Hint I can give you:

Hint 1.a

You probably found it, this puzzle is to find a certain $f$, which takes in input a number and returns one of the $26$ letters of the alphabet. This hint is to give some facts on $f$ and its link to this puzzle.
1. You also probably know that but the aim of this puzzle is to find $f(19)$, $f(8)$, $f(7)$ and $f(10)$.
2. $f$ is defined on any positive natural number, it is not defined on zero, it is not defined on negative numbers. It could be possible to define it by being treacherous but I haven't done it and that's not necessary for the puzzle. Plus there is no minus symbol in the numeric keypad 😊!

Hint 1.b

As it's defined on positive numbers, here is a list of its values:
$f(1) = p$
$f(2) = y$
$f(3) = u$
$f(4) = v$
$f(5) = k$
$f(6) = d$
$f(7) = ?$
$f(8) = ?$
$f(9) = w$
$f(10) = ?$
$f(11) = p$
$f(42) = e$

Please specify in your answer whether you solved the puzzle with or without hint 2.a and hint 2.b

Hint 2.a 7 July 19

After being stuck for hours within this cave, my sister noticed that the ground was discreetly etched by the 26 letters of the alphabet. There was a mathematical symbol next to the 26 letters.

Hint 2.b 14 July 19

The mathematical symbol is $\%$.

Hint 3.a 17 July 19

Actually modulo operation is used once or twice, or isn't used, to encrypt each number. Another important operator used within the encryption is $+$, it's systematically used once. The latter could also have been etched to the ground!

Hint 3.b 17 July 19

Hint 2.a is as much important as Hint 2.b. The fact that the alphabet is etched is important.

Answer 1

The answer is:

She pressed the Gold button.

How the transformation function works:

Take the $n-th$ letter of the number $n$ when written out in English and shift this by $n$ positions. If a number consists of less than $n$ characters, restart from the beginning. If by shifting a letter by $n$ positions we reach the end of the alphabet, again, we restart from the beginning.

So for instance:
$1 >$ ONE $>$ P (O shifted by $1$ position)
$2 >$ TWO $>$ Y (W shifted by $2$ positions)
$3 >$ THREE $>$ U (R shifted by $3$ positions)
$4 >$ FOUR $>$ V (R shifted by $4$ positions)
$5 >$ FIVE $>$ K (F shifted by $5$ positions)

For numbers $19, 8, 7, 10$ we obtain:
$19 >$ NINETEEN $>$ G (N shifted by $19$ positions)
$8 >$ EIGHT $>$ O (G shifted by $8$ positions)
$7 >$ SEVEN $>$ L (E shifted by $7$ positions)
$10 >$ TEN $>$ D (T shifted by $10$ positions)

This leads us to the correct answer: GOLD, so the gold button must be pressed.

Quite challenging, but very nice puzzle!

Previous answer below:

Alright, nothing close to a solution yet, but I'll just post some of my observations here and maybe it helps someone else solve it:

With the numeric keypad some input number may be supplied, which will then be transformed, via a function $f$, to one of the $26$ letters of the alphabet.
The blue screen shows the result of this transformation, once a number is supplied.
The red screen, as mentioned in the comments, is simply a countdown, showing the number of attempts left.
The five buttons are colored buff, gold, corn, opal and puce. The right button must be pressed to leave the cave.

Initially, the blue screen shows the numbers $19-7-8-10$. If these numbers are given as input, no result shows up on the blue screen, probably because this is the secret code we need to guess.
Since all the colors consist of four letters, the transformation on these four numbers probably leads to one of the five colors, i.e.: $f(19)-f(7)-f(8)-f(10) = b-u-f-f$ OR $c-o-r-n$ OR $g-o-l-d$ OR $o-p-a-l$ OR $p-u-c-e$

OP hinted that the transformation is somehow related to the mathematical modulo operation, which gives the remainder after division of two numbers.
OP further remarks that all mathematical symbols are pivoted by 45 degrees, meaning e.g. that x becomes + and vice versa, although it is unknown which, if any, of these operations are used in the transformation function.

Then there is the question of how a transformation on a number would lead to a letter. This might be where the modulo operation comes in: if $v$ is the numeric result of the function applied to the number $n$, then $v%26$ will lead to a number in the range $[0,25]$, which can easily be mapped to a letter, e.g. $a>0$, $b>1$, ... Unfortunately, this sheds no light on how to obtain this numeric result from the original input number...

Looking at the list of values as supplied by OP, we can see that both the number $1$ and the number $11$ are transformed into the letter $p$. Incidentally, this corresponds to the first letters of the $1st$ and $11th$ word of the message: "Press the wrong button and that'll be your last action, press the right button and leave this cave."
The only other double in the list is $f(4)=f(17)=v$, which may suggest that the initial sequence of $p, y, u, v$ is repeated, with $f(13)=y$ and $f(15)=u$. But since $f(19)$ would then have to yield $k$ we can see that this does not work, because none of the colors start with the letter $k$.