How do I create a fast algorithm for distributing 2000 cards among 1000 people With 2000 index cards, each having a single number on it, I am tasked with the job of finding the grand total of all numbers. In order to do so, I have 1000 people, each who sit at a desk in a room that has 25 rows with 40 desks. Each person can pass a stack of cards to their neighbor front, back, right and left. Given that I am only allowed to give the entire stack to one person, it takes 3 seconds for a person to pass a stack (regardless of size) to their neighbor. It takes 1 second to add 2 numbers and the person allowed to write the sum on the card to pass to the neighbor.
What's the fastest means of obtaining the sum of all numbers on the card if I am required to gather all subtotals to give to the original person I passed the entire stack to?
I'm wondering whether it's fastest to utilize all 1000 or whether there's a faster way of getting the sum without having to use all. 
EDIT: I feel like the best thing to do is to give the initial stack to the person in the middle so that it'll be quickest to have everything passed back to him/her
 A: I don't know if this is optimal, but here is a pattern to beat:
Give the whole stack to the person at $(2,2)$. They then pass to the person at $(2,3)$, and then both pass to $(3,2)$ and $(3,3)$. The four then pass to $(1,2), (1,3), (4,2), (4,3)$ and then the eight pass so that everyone in the block from $(1,1)$ to $(4,4)$. Now, $16$ people have cards, and we are $12$ seconds in. Group the people in terms of the round in which they received cards - there are groups $P_0, P_1, P_2, P_3, P_4$ with $1,1,2,4,8$ people respectively.
$P_0$ takes $136$ cards, $P_1$ takes $132$ cards, each person in $P_2$ takes $128$ cards, each in $P_3$ takes $125$ cards, and each in $P_4$ takes $122$ cards. Silence reigns for the next $121$ seconds as everyone sums cards. 
Then, everyone in $P_4$ is done and passes their cards to their neighbor in $P_3$. This takes $3$ more seconds, while everyone else keeps working. Then group $P_3$ finish their cards, having taken $125$ seconds, including adding $P_4$'s sums to theirs. They then pass to $P_2$, who are waiting and so on.
At the end, this takes $137 + 12 = 149$ seconds for $P_0$ to have the total.
For a loose lower bound, note that someone needs to sum all $2000$ cards, so we can't do better than (around) $\min_k \frac{2000}{2^k} + 6k$, where $k$ is the number of rounds of passing, overestimating the number of people after $k$ rounds as $2^k$. Wolframalpha gives that bound at around $55$ seconds.
A: Following on Michael Biro's suggestion, give the stack to somebody in the center and call that position $(0,0)$ and that person $P_0$.  They pass cards to $P_1$ in $(0,-1)$.  Each of them next pass to $P_2's$ in $(1,0)$ and $(0,-2)$.  The third round the first two pass to $(-1,0)$ and $(-1,-1)$ and the $P_2$s pass to $(1,1)$ and $(1,-2)$.  These are $P_3$s.  Everybody can pass to a $P_4$, but $P_0, P_1$ and two of the $P_3$s cannot pass to a $P_5$.  Everybody else does so.  We now have one $P_0$, one $P_1$, two $P_2$s, four $P_3$s eight $P_4$s and twelve $P_5$s.  $P_0$ keeps $81$ cards, $P_1$ keeps $79, P_2$ keeps $77, P_3$s who pass to $P_5$s keep $74, P_3$'s who don't pass to $P_5$s keep $77, P_4$'s keep $71, $ and $P_5$'s keep $68.$  It takes $15$ seconds to pass cards, $68$ for the $P_5$s to add, $15$ seconds to pass the cards back for a total of $98$ seconds.  There may be an additional $5$ seconds if you need a second to add the number from te stack you received.  
We can do a little better passing $6$ times.  We can only get cards to $1$ people in round $6$, but the $P_6$s can do $42$ cards.  We have $18$ seconds passing, $42$ seconds adding, $18$ seconds passing back for a total of $78$ seconds.  Even better we can enlist $20\ P_7$s to do $28$ cards each, for a total time of $21+28+21=70$ seconds.
