Problem solving relies on what came before and what will come after

Sometimes it is not just what you do but the order in which you do it that matters. You absolutely must develop an ability to see several layers of consequences ahead of the solution as well as several layers of cause and effect behind the problem and then juggle them at once in the way that a chess player does. Otherwise you will be easily herded into disasters by cleverer strategists with good advisers and teams who pay more attention. Below is a great classic example that many will be familiar with.

How to solve the Fox the Goose and the Corn problem

Constraints:

  1. Fox and goose can’t be left together.
  2. Goose and corn can’t be left together.
  3. Only one can travel with the man in the boat so two must be together at some point.
  4. It will take at least 3 boat trips to carry them one at a time.
  5. Options

Fox and goose                   no

Fox and corn                      yes

Goose and corn                no

The logical method

The Goose is in both prohibited pairings therefore she can’t be left on either side. Only Fox and corn can be left together without the man present. It seems likely that keeping the goose close might help. Let’s try.

The key to these puzzles is to be able to switch constantly between the helicopter view and the detail view. The Big picture is that we want to get all this to the other side.
The small picture view is that we want to keep the goose alive and prevent her eating the corn while we get them all across. If we are too focused on the big picture we will build a mental barrier that says we can’t bring stuff back, what would be the point.
The small picture view says we do whatever is needed to keep goose alive and corn uneaten.
Small picture view also says that that each step is just a single piece of the solution not the whole thing.

 

Solution
(means in transit)                                                           Left side                                 Right side

Man Fox Goose Corn

Take the goose across

Then get the fox but bring the goose back   now     Corn               (Goose)          Fox

Now leave the goose and bring the corn                  Goose             (Corn)             Fox

Now return and get the goose                                                            (Goose)          Fox Corn

Result                                                                                                                                 Man Fox Goose Corn

——————————————————————

By extracting the rules from constraints and options, we were able to steer fairly easily
through a complex problem and find the first of only two feasible solutions. Open mindedness, I.E. working without false constraints, made it possible to spot the obvious move that most people miss.

 

 

Using a systematic approach

Create all the possible configurations and a node for each
Create a node for each boat trip

Since a boat trip can only move one item between nodes via a boat trip

If we allow GF or GC to exist without M the game is over

M =Man  G=Goose F=Fox C=Corn

 

1 MGFC –  2 g FC – 3 mgf – 4 mgc – 5 mg – 6 mf – 7 mc – 8gf – 9 gc -10 fc =11 0

Now arrange these nodes so none of the forbidden combinations can exist

1 MGFC – 3mgf – 4mgc – 5mg – 6mf – 7mc – 10fc   allowable combinations

5mg – 6mf – 7mc      allowable boat trips

 

Action                outcome 1             outcome 2             comment

MG                         FC                               G

MF                            C                                F                      goose in transit

G                               FC                     bring back goose

MG                                                         MGFC

 

Using Graph theory

Here we focus only on the combinations or “states” that don’t cause the illegal pairs to be left together and we create a node for each. Below we show them numbered 1 to 10. The first is Man, Fox, Goose, Corn together on one bank.

1:(M,F,G,C)    2:(M,F;G) 3:(M,F,C) 4:(M,G;C)  5:(M,G)  6:(F,C)   7:(F)  8:(G)  9:(C)  10:(0)

e.g MC would mean that Fox and Goose were left behind, not a good outcome

In this series of nodes, 1 describes the initial state where all four are on one bank and 10 describes the goal state where all four are on the opposite bank.

 

What we want to do now is find the edges (links or journeys) between nodes which could be achieved by one of the allowed boat trips.
e.g
1 -6 is allowable because FC would be the outcome of boat trip MG therefore legitimate.

That leads to the graph below which shows that in fact there are two legitimate solutions to the problem.

7–2

/          \

1–6–3     8–5–10

\          /

9–4