42 and the missing 5 or How to see Patterns

The purpose behind this essay is to give an idea of what to expect when tackling a harder than normal problem and is slightly tongue in cheek.

Step one is obviously, find a problem hard enough to be a challenge, it doesn't matter if it is very hard. You may choose a problem with an already known answer and therefore be able to receive help. The benefits are quicker solution, but the drawbacks will be less experience overcoming difficulties. I've never known mathematicians scoff at another's inability to cope with basic ideas, but if you are proud and confident that with practice improvement will follow, why invite mockery. Choosing a problem they can't do guarantees problem solving experience comfortable in the knowledge no-one is going to say your doing that wrong and here is the answer.

Surprisingly it's not hard to find a hard question in mathematics, Number Theory dealing only in whole numbers is full of them, with the added advantage they are easy to state.

One specific problem is "The 3n+1 Conjecture",_{[Proposed by Professor Brian Thwaites et al]} references in the tab for edited entry, states; starting with any positive whole number, generate a new number by the rule, if it is odd multiply it by 3 and add 1, and otherwise divide by two. Applying this rule to every generated new number you get a sequence or collection of numbers terminating in 1. Example; starting with 7 generates 3x7+1=22, divide by 2 gives 11, and so on giving, 7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1, done!

At first their is a feeling of exhilaration with the prospect of solving such a hard problem, but that soon succumbs to feelings of inadequacy, staring at space without a clue how to proceed. It is a good idea to write something down in order to try and understand the problem, make it your own and get close to it. In this case lists of completed sequences might be helpful, try to calculate them without the use of a calculator because the mental practice is needed to sharpen the mind, and it helps block out distraction, because you are going to have to get up and very close to this problem. Distraction is your enemy stay focused.

Of course writing down lists on their own will not solve the
problem, but at least it is something to look at, and you should be looking for some sort of pattern, it may even be useful to reformulate the problem into something more familiar.

It is important to read, to get away from the problem and avoid fixation. Ask questions about what you read and make sure you understand the answers, because they may be useful. Don't be embarrassed on finding the answers show you have missed the obvious, but remember what is important is your continual improvement and you therefore will be better placed to ask good questions and importantly, solve the problem.

Many setbacks will follow, you may have noticed the algorithm can be processed by a computer, and by reading and asking questions learn about Turing_[1912-1954] and this thing called a Halting problem_[1937],loosely meaning computers will


never be able to decide if the sequence halts or not. With humility you may then decide that without evidence computers are and always will be inferior, to err on the side of caution, and look for another approach not explicitly using the algorithm. This may mean the complete abandonment of your original ideas and starting again.

Different approaches lead to different difficulties and be prepared to be forced to change approach several times, you will then be experiencing the same roller coaster feelings that the greatest exponents in your field experience everyday. But it is not as important to solve the problem, as gain from the experience tackling it, with positives of focused reading, better directed questioning and the lateral thought_{[Edward DeBono]} experience in your problem solving skills.

Eventually the problem will be stored in the back of the mind or solved, but you will be better equipped for the next challenge.

Now about 42 and the missing 5; 42 is the least non-trivial (not of the form 2ⁿ) even whole number that doesn't generate a sequence with a 5 in it.

Some useful references are included, a free course in how to study, some very interesting sites related to Turing, a lecture about undecidability that's not to technical, and ways to think out of the box.