Lesson 7.02: Sequence Generation & Particular Solutions

This approach is particularly useful for identifying patterns in problems involving numerous objects.

Sequence generation works by simplifying the problem to its basic components: the input and the output. By focusing on these elements, we can set aside the distracting details and concentrate on finding patterns. Starting with the simplest cases, we build up our understanding and can eventually predict more complex scenarios. By mastering sequence generation, you'll enhance your ability to solve complex problems efficiently and accurately.

Identifying when a relationship within a problem is likely to be linear is important. The linear relationship makes it easy to predict the output for any given input, and appreciating this allows you to make the most use of the sequence-generation technique, especially when faced with time pressures.

Sample Question

Try this problem yourself, before reading the analysis

A set designer is constructing a stepped platform for a choir to stand on. The platform will be

constructed of cubic boxes, arranged to form steps. A section of the platform, on which five

rows of choristers could stand, four on each row, is shown on the right.

The set designer needs to ensure that all the parts that are visible to the audience before the choir come in are covered in black felt. The flat sides (with 15 squares shown on the diagram) will be visible, as well as the fronts and tops of the rows where the choir will stand. The faces which stand against the back wall and on the floor will NOT be visible.

If she only has 100 felt squares, and she covers the largest 5-row platform possible, what sized choir could it support?
Find the number of felt squares needed to cover the platform if it is to be capable of supporting 5 rows with 50 choristers on each.

An appropriate input and output are fairly easy to spot here: the number in each row (input) and the number of squares of felt needed (output).

A table allows you to concentrate on each case one by one, and to store them to review later.

It is easy to see that this sequence leads to 7 in each row, needing 100 squares. This means (7x5) = 35 choristers.

In this question it is necessary to state the relationship between the number in each row and the number of squares formally. Because each person per row adds on 10 more squares, and the end pieces require 15 squares each, the following formula is reached:

(number of people per row x 10) + 30 = total number needed.

If there are 50 people in a row, then that will require (50 x 10) + 30 = 530 felt squares.

When faced with a problem that covers a wide range of possibilities, it can be helpful to focus on a single, specific example. By choosing one arbitrary value to represent all possible values, you can simplify the problem and make it more manageable. This approach allows you to gain insights and find solutions without getting lost in the complexity of the general case.

By applying the 'particular solutions' strategy, you'll be able to navigate and solve even the most disorienting problems with greater ease.

In some cases where a general problem is considered, choosing an arbitrary value, or set of values, does not shed much light on the problem. In problems like this it is worth considering what an extreme case is for one of the variables, since this may make the logic of the problem more stark. One example is to consider what happens when a variable equals zero (if that is a permissible value) since that can reveal how the other variables are related. Alternatively, considering the largest possible value can allow its influence on other variables to become clearer.

if the claim is true generally
then it is true for all values of the key variable
and then it is true for any values
so you can choose whatever you like for the variable
so you can choose the most extreme value.

Sample Problem

Try this problem yourself, before reading the analysis

airplane flies in a straight line from airport A to airport 13, then back in a straight line from B to A. It travels with constant engine speed, and there is no wind. Will its travel time for the same round trip be longer, shorter or the same if there is a constant wind which blows from A to B during the whole return journey?

This question has the tell-tale lack of detail which often allows for a particular solutions approach. Implied in the question is the claim that the travel time will be longer, shorter or the same regardless of what the engine speed is, and what the wind speed is.

One could make up some simple numbers for these, for example: the plane goes at 100 km/h and the wind blows at 10 km/h.

But this still leaves us with some careful analysis of distances, speeds and times to perform.

If, following the reasoning discussed previously, it is true for any speeds, then we should consider extreme speeds, to see if this makes the problem simpler. What is the extreme case, in terms of movement? Well, an extreme would be for the wind to be going the same speed as the plane. In that case, the plane will go fast in one direction but, in the other direction, the speed of the wind will cancel out the speed of the plane - so the plane will go nowhere! As a result, the round trip will never be completed. This result shows that a round trip with a constant wind will definitely take longer than a round trip with no wind.

Particular solutions are possible to introduce only when you are being asked to find a solution that is generalised and which applies over a range of values. These types of problem are often daunting because they are general, and there is little that is concrete enough to diagram or tabulate or list.

When choosing a particular input you should either aim for numbers that are 'easy', or ones that are in some sense 'extreme' (therefore making the logic of the question easier).

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