I have some questions regarding the Invariance Principle commonly used in contest math. It is well known that even though invariants can make problems easier to solve, finding invariants can be really, really hard. There is this problem from Arthur Engel's book titled "Problem Solving Strategies" which states:

In a regular hexagon all diagonals are drawn. Initially each vertex and each point of intersection of the diagonals is labeled by the number 1. In one step it is permitted to change the signs of all numbers of a side or diagonal. Is it possible to change the signs of all labels to −1 by a sequence of steps?

We are supposed to use invariance in this problem. I am very bad at locating the invariant unless it is something as simple as modn of some value in the problem for a particular value of n. I could not solve this problem using an invariant; however, I solved it by arguing that if the center point S of the attached figure carries number -1 along with the points enter image description here N, H, Q and L, then G will be forced to be +1. I am not elaborating on the complete solution, as that is a distraction from my main question.

Engel's solution feels like she or he is a genius who came up with it. Referring to the figure again, we can see that the product of the nine numbers on M, H, N, O, J, P, Q, L, and R does not change (another figure shown below to picture it better).

enter image description here

Everytime I managed to solve an invariance-based problem, I solved it by not using Invariance Principle. I have two questions.

  1. How do you get better at applying the Invariance Principle? I have solved a lot of problems and managed to solve ... maybe only 60% of what I attempted. I want to reach at least 80 to 90% of success rate.

  2. Are there other good sources for invariance-based problems? I am almost done with Arthur Engel's "Problem Solving Strategies" and I am completely done with Paul Zeitz's "The Art and Craft of Problem Solving" as far as invariance chapter is concerned.

Invariance has been one of my biggest handicaps in problem solving. I would greatly appreciate it if you can please answer the two questions above.

  • $\begingroup$ My biggest handicaps so far so is Invariance and Functional Equations.I need help and I mean it. $\endgroup$ – Icosahedron Aug 8 '17 at 21:19
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    $\begingroup$ @Icosahedron I am also in the same boat. For functional equations, there are books dealing with it. There are also some nice handouts available. But invariance is a beast and I have not found any good books on that topic. I wish someone on this forum would recommend a good source of problems (at the level of IMO/Putnam) for Invariance. $\endgroup$ – TryingHardToBecomeAGoodPrSlvr Aug 8 '17 at 22:35
  • $\begingroup$ I wish you luck buddy, I have Engel too here, I understand the simple problems but when asked to solve a difficult problem, I end up solving it without invariance just like you said. I'll mention you in a thread whenever I see a good textbook devoted to this topic, also, can you share me some useful resources on Functional equations? $\endgroup$ – Icosahedron Aug 8 '17 at 22:55
  • $\begingroup$ There are a lot of books and resources for functional equations. I will mention a few here. Evan Chen's handout -- link: web.evanchen.cc/handouts/FuncEq-Intro/FuncEq-Intro.pdf, B. J. Venkatachala's book -- B. J. Venkatachala. Functional Equations: A Problem Solving Approach -- Amazon link: amazon.in/Functional-Equations-Problem-Solving-Approach/dp/…, Titu Andreescu's book titled Topics in Functional Equations -- Amazon link: amazon.com/Topics-Functional-Equations-Titu-Andreescu/dp/…. Hope this helps. $\endgroup$ – TryingHardToBecomeAGoodPrSlvr Aug 9 '17 at 17:13
  • $\begingroup$ Speaking of the "product" here makes it sounds more clever than necessary. It is just a fancy way of saying "+1 if an even number of them are -1, and -1 otherwise", i.e describing the parity of the number that are -1. The nine chosen points have the property that when you change all signs on a side or diagonal, you change an even number of those chosen points (zero for a side, two for a diagonal). So in all cases the number of -1 chosen points has its parity unchanged - it remains even. 9 is not even, so they can never all become -1. $\endgroup$ – Nate Eldredge Aug 13 '17 at 6:09

Being officially a guy which your are trying hard to become, :-) answering your question I feel obliged to tell you the truth: :-) experience is the best of the teachers. :-) Sometimes you may find a problem solver handbook, helpful for a specific topic, but it won’t be solving problems for you. :-)

Concerning your explicit questions.

1 In order to get better in problem solving matters not the success rate, but level, hardness, depth of problems. Solving thousands of quadratic equations won’t make anybody a good problem solver. :-) On the other hand, I’m solving open problems, so any success rate greater than zero is good for me (then I may publish my solution and to show my boss for what I’m paid :-)).

Of course, IMO level problems are pretty hard. Nevertheless, they should be solved in hours, whereas really hard problems may be solved during decades. My MSE examples: of school level and of professional level). Also parts of this answer may be relevant.

2 Unfortunately, my problem solver handbook knowledge is rather narrow, because I have to solve problems, but not to read how to do this. :-) Also a lot of my problem solving experience is related with old Russian school sources.

As a general problem solver handbooks I may highly recommend George Polya books, especially “Mathematical discovery: on understanding, learning and teaching”. For specific subjects may be useful collections of olympic problems with solutions, grouped into topics. Also you can use tag search at MSE.

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From what I hear jj sylvester pretty much invented invariant theory. Maybe take a look at some of his work. He was apparently a pretty good mathematician. .. well, upon looking it up it appears Cayley gets the credit: the idea was suggested to him by an elegant paper of Boole. Here is the wiki. Looking up sylvester, he made fundamental contributions. .. Then again, Encyclopedia brittanica gives credit to both .

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