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4.5
Along with J Mason's "Thinking Mathematically," this is the only book which helps young (aspiring) mathematicians to understand how someone manages to "Pull the Rabbit out of a Hat" (famous Deus Ex Machina expression mentioned by Polya in one of his very good books on problem solving). Since childhood, I have been wondering, just like most of the people I meet, how smart (even geniuses you might think) people solve certain problems whose crux move ("crux move" as described by Paul Zeitz the great in his book "Art and Craft of Problem Solving" as the most critical insight in the solution of a problem) is impossible to discover (atleast by ordinary mortals like myself).****************************************************************There is only one solution to 7^x-3^y=4 namely x=1, y=1. Why?Problems like these involve some observation of the "Mathematical Situation" which has to be figured out using the givens from the problem and also from other facts already known from basic math.7^x-3^y = 4. Let us call Sx the least difference in the set {7^x-3^y} for number x. For x=1, Sx=4. For x=2, it is 22. etc.x=1 and y=1 is a solution by inspection. Let x be greater than or equal to 2. Hence y is greater than or equal to 2. For x=2, Sx=22 and y=3.1. If Sx=4, then x and y must be of the same parity. This can be observed by looking at the residue of 7^x and 3^y mod4. They both alternatebetween 1 and 3 mod4, according as x=even or odd respectively.2. Sx (mod 9) = 4 if Sx = 4 for some x. Since 7^x follows a repeated pattern mod9 (7 4 1 7 4 1....), we can infer that x=3k+2 meaning x=2 mod3.3. Sx (mod 7) = 3 ( Sx (mod7) = 3 since -3^y appears in Sx with a negative sign). Similar to 2, we can infer that y=6l+1 meaning y=1 mod6. Notethat this also implies y is odd.4. Since x and y are of the same parity (from 1), we can infer that x=6k+5 (since y is odd, x must be odd) meaning x=5 mod6.5. 7^x-7=3^y-3. Hence if d|x-1, then 7^d-1 divides 3^y-3. Similarly if d|y-1, then 3^d-1 divides 7^x-7.6. From 4, 2 divides x-1. Hence 7^2-1 divides 3^y-3. 7^2-1=48=3*16. Hence 16 divides 3^(y-1)-1 meaning 3^(y-1)=1 (mod 16) meaning y=1 mod4( this can be inferred by looking at the pattern formed by 3^(y-1) mod 16 as y is varied. It repeats (3 9 11 1 3 9 11 1....) every 4th value in y-1).Together with the result in 2. above, we can infer that y=12k+1 or y=1 mod 127. From 6. above, we can deduce that 3^12-1 divides 7^x-7 since 12 divides y-1. Since 13 is a factor of 3^12-1, we can deduce that 13 divides 7^x-7.Now (7,13)=1. Hence 13 divides 7^(x-1)-1. Now 7^(x-1) mod 13 repeats with a period 12 with 7^12=1 mod13. Hence x-1 is a multiple of 12 or x=1 mod 12.At this point, we have a contradiction. The point number 4. above shows that x=5 or 11 mod 12 while point number 7. above shows that x=1 mod 12.Both can't be true simultaneously. Hence the initial assumption that Sx=4 for some x must be false.****************************************************************There were times when I could not solve such problems. When looking at the solution, I would think 'how did he even think of this fact' or 'what is the way to go about figuring out such facts?' and the answer is that it is through systematic and intelligent investigation. I learnt about math investigation through Gardiner's book. I learnt that solving math problems is like putting the pieces of a puzzle in intelligent ways to obtain a proof. The only difference between jigsaw puzzles and math is that the pieces have to be developed here! We are not given the pieces.Just imagine. If they had given all the pieces of the proof and we just had to use them, would it not make the problem very easy to solve? Gardiner does a great job at describing the ways to form the pieces of the puzzle needed to solve the problem.Difficult problems are difficult because the fact that connects the givens to the fact that is needed to be proved (or the object that needs to be found) is not at all obvious. It has to be discovered or known from past experiences. This book and Mason's book are probably the ONLY books that attempt to provide guidance to the aspiring mathematicians as to how such facts are discovered. Beyond that, this book also helps in doing math investigation on open problems which are not tough contest problems but are open ended or are unsolved problems.He takes many examples and guides the readers through the process of investigation on very interesting problems. The final two examples he takes are really very difficult for someone who is new to mathematical investigation. I bet that this book will be an eye opener for people who are trying hard to get through contests or for people who are trying to do research level math on their own or with little guidance from their prof. One of the problems he takes for investigation is the postage stamp problem which is really hard to investigate and Gardiner has done a great job at walking the reader through the entire process of investigation. He has also shown ways to see insights and the ways of thinking needed to become a good problem solver.Gardiner has written a chapter in "The Princeton Companion for Mathematics" where he says"Attempts to teach "problem-solving" in schools often misconstrue mathematics as a kind of "subjective pattern-spotting." Instead of correcting this distortion at university level, mathematicians often maintain a discreet public silence about the very private matter of how serious mathematical problems actually get solved."He is absolutely right! Where do you find mathematicians describing the method through which they discovered or solved tough problems? EVERY contest problem solving book gives the problem and the solution to it. No pointer to the ways to discover the insightful fact needed to solve the problem. It strongly points to the possibility that they do not want to let it out. I have spoken to really good mathematicians and it is true that very few actually speak about how they discovered the solution to the problems that they solved.It is also possible that they are not conscious of their thought process.Either way, there is no real way of learning the systematic digging needed to solve really tough contest problems of Math Olympiad and Putnam. This is one of the very few books (along with Mason's book and Paul Zeitz's book) which is extremely useful in learning the skill of "Mathematical Digging" needed to solve tough contest problems and to learn the ways to break into the mystery of "Deus Ex Machina".
