Post Number:#3 by Thamim Zahin » Tue Jan 31, 2017 9:05 pm
Let $N = (d_3,d_2,d_1,d_0)$ mean $N = \overline{d_3d_2d_1d_0}$. For $P = (a,b,c,d)$ with $1 \leq a \leq b \leq c \leq d$ we consider the following three cases:
Case 1: $d=a$. Then $a=b=c=d$, yielding $Q=P$, which means $X = P - Q = 0$ and $Y=0$. Hence $X + Y = 0$.
Case 2: $d>a$ and $c=b$. Then
$X = Q - P = (d,b,b,a) - (a,b,b,d) = (d - a - 1,9,9,10 + a - d)$.
If $d = a+1$, then
$X + Y = 999 + 999 = 1998$,
and if $d > a+1$, then
$X + Y = (d - a - 1,9,9,10 + a - d) + (10 + a - d,9,9,d - a - 1) = (10,9,8,9)$.
In other words, $X + Y \in \{1998,10989\}$.
Case 3: $d>a$ and $c>b$. Then
$X = Q - P = (d,c,b,a) - (a,b,c,d) = (d - a, c - b - 1, b - c + 9, 10 + a - d)$,
yielding
$X + Y = (d - a, c - b - 1, b - c + 9, 10 + a - d) + (10 + a - d, b - c + 9, c - b - 1, d - a) = (10,8,9,0)$.
In other words, $X + Y = 10890$.
Conclusion:The possible values of $X+Y$ are $0, 1998, 10890,10989$.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.