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This essential trains for: SAT-I, GMAT, AMC-8, Math Kangaroo 5-6, Math Kangaroo 7-8.

Counting objects on a line: If we place N objects on a line then there are N-1 "spaces" between them. (This is true for any open (not closed) contour, not necessarily a line.)

Though simple, this concept is often deceptive in problems.

Example:

In an orchard, there are 81 persimmon trees planted in a square grid pattern. At the center of each group of 4 trees there is a small service station. How many service stations are there?

There are 9 trees on each row and column. Between the 9 trees there are 8 gaps. There are 64 service stations.

Obviously, if the objects are on a closed contour there are only N gaps between the N objects.

Example:

Is a restaurant manager right in assuming that, by placing a dinner candle between each pair of seats around a table instead of in front of each plate, the restaurant cuts on expenses?

No. They will use the same number of candles in both situations because the table is closed contour (whether it is round or rectangular.)

How many objects are there between two ordinals?

Between N-th and M-th there are:

excluding the endsequation
including one endequation
including both endsequation

figure

Example:

If Papa Smurf is paying for his groceries at the cash register and Handy Smurf is 11-th in line, how many Smurfs are standing between Papa and Handy? (characters from Peyo's "Smurfs")

There are 9 Smurfs between Papa and Handy.

Worst case analysis: Pilot Pirx's space solitude is disturbed by an asteroid on a collision course. The ship is equipped with 10 long range missiles, 5 4-pi steradian clusters and 2 asteroid annihilators. Pilot Pirx suffers from deep space attention deficit and has no idea which is which. What is the maximum number of pieces of ammunition Pilot Pirx may have to waste until he is sure to have fired an asteroid annihilator?

Obviously, Pirx could be very, very unlucky and have to fire 15 missiles before he gets to a useful one.

(characters from Stanislaw Lem's "Tales of Pirx the Pilot")