How to find the highest sums from several groups of numbers What is the equation or a way to figure out what the highest possible sum is from several group of numbers.
For example, If i had 4 groups of numbers, and in each group the numbers 0-10 occur. how can you add those numbers up to the highest outcomes. an easy way of explaining is in a car race. There are 4 races, and 10 participents in the race. Winning the race gives 10 points, second gives 9 and so on. What would the highest possible score be for someone to have and yet still be the losing racer. Is there a concrete equation for this problem that works for any set of numbers?
 A: In your example of $c=10$ competitiors and $r=4$ races with points $c=10,9,\ldots,1$ each race, you can say:

*

*$\frac12c(c+1)=55$ points are distributed in each race

*$\frac12rc(c+1)=220$ points are distributed in total

*The average number of points per racer in total is therefore $\frac12r(c+1)=22$

*It is possible in this case for one racer to get $\frac12r(c+1)+1=23$ points in total and all the others $22$ except for one who gets $21$.  In some other competitions (an even number of racers and an odd number of races) to have a clear overall winner might require them to have a total $1.5$ points above average.  So we could say $\big\lceil \frac12r(c+1)+1\big\rceil$ as the minimum number of points needed to be a clear winner in general

*The lowest second place total which not tied with others is $\big\lceil \frac12rc+1\big\rceil=21$, essentially the same as the previous result but ignoring the top place winner of every race.  The lowest second place total which might be tied with lower positions is $\big\lceil \frac12rc\big\rceil =20$ here.

*The highest total possible points is $rc=40$ points here.  If somebody scores this then the next highest possible is $r(c-1)=36$, but it is possible to narrow this gap by transferring points between the top two overall until they are almost equal.

*The highest second place total here must be strictly below $r\left(c-\frac12\right)=38$ if it must be strictly less than the points won by the  top overall place, so cannot be higher than $37$ here.  Let's call this $$\bigg\lfloor r\left(c-\frac12\right) -\frac12\bigg\rfloor$$ as the general expression for the highest possible score for somebody clearly second top overall.

