Last updated: Feb 24, 2024

Solving Mathematical Problems (Second edition)
Terence Tao
Oxford University Press, Oxford, England: 2006
Paper, 128 pages. ISBN13: 9780199205608
ISBN10: 0199205604

This book discusses various Olympiad level problems and how one can go about trying to solve them. It is the second edition of an earlier first-edition run.  It has also been translated into Portugese as “Como resolver problemas matemáticos” by Paulo Ventura Araújo for the Sociedade Portuguesa de Matemática; into Chinese as “解题·成长· 快乐——陶哲轩教你学数学” (ISBN 9787301154472) by  Qinglin Yu (于青林) for Peking University Press; and into Italian as “Risolvere problemi matematici. Il mio punto di vista” (ISBN 8896973880) by Samuele Maschio amd Carlo Càssola for Scienza Express Edizioni SNC (reviewed here by Alberto Saracco).

It has been reviewed for the Notices of the American Mathematical Society by Loren Larson.

— Errata —

  • In page 2, “Problem 1.1 question” should just be “Problem 1.1”.
  • In page 6, second to last paragraph, “once can avoid” should be “one can avoid”.
  • In page 7, “ompute” should be “compute”, and “put clear denominators” should be “clear denominators”. On the fourth displayed equation, 4d^2 should be 4d^4, and 196t^2 should be 192t^2.
  • In page 9, example (e), “876” should be “376”.  “which are exactly” should be “which have exactly”.  In example (d), n+1 should be n(n+1)/2.
  • In page 16, bottom, “217” should be “2^{17}“.  “if this question is true, then the original question is true” should be “if the answer to the original question is “yes”, then so is the answer to this question”.
  • In page 25, third paragraph: one of the “n”s should be in math mode.
  • In page 33, Exercise 2.5: For an additional challenge, prove this exercise without using Bertrand’s postulate.
  • In page 35: In the quote, “that was originally” should be “than was originally”.
  • In page 37, second display: f(f(2)-1) should be f(f(f(2)-1))).
  • In page 40, “smells heavily on” should be “smells heavily of”.
  • In page 42, t^m should be t^n in one of the displays.
  • In page 44, 5ab should be (13) (two occurrences)
  • In page 45, Problem 3.4, there should be no commas between (x-a_1)^2 and (x-a_n)^2. “are all integers” should be “are all distinct integers”. One should delete all references to a_0, for instance deleting the factor (x-a_0)^2 in the problem box, and also p(a_0) and q(a_0) in the next page.
  • In page 46, in the sentence “But polynomials only have as many degrees of freedom as their degree”, insert “with leading coefficient 1” after “polynomials”.
  • In page 47, Exercise 3.7, the +1 should be a -1, one should look at p(x)+q(x) rather than p(x)-q(x), and “are integers” should be “are distinct integers”.
  • On page 50, the intersection of BC and AI should be labeled D.
  • In the second to last line on page 52, “either \beta = 60^\circ or \alpha = \gamma” should be “either \alpha = 60^\circ or \beta = \gamma“.
  • In the diagram on page 53, the angle at D should be \gamma + \beta/2, and the angle at E should be \beta + \gamma/2.
  • In page 58, one of the instances of ABEF should be in math mode (like all the other instances).  “on the same side of AB” should be “on the same side of AC“.
  • In page 63, third line, “inner square” should be “inner rectangle”.
  • In page 65-66, the informal geometric argument given is incomplete, the issue being that just because the sum of side lengths of R_1 is (say) bigger than 1, it is not immediately obvious that the same is true for, say, R_2. But one can check using algebra that if a+b > 1, then (1-a) + \frac{ab}{1-a} > 1, and similarly with the inequalities reversed; this allows the argument as stated to be made rigorous. (One can also argue by considering the rectangle with the narrowest side, and showing that it is adjacent to one which is even narrower if its sides do not add up to length 1.)
  • In page 66, problem 44, there is a =2 missing at the end of the string of equations, thus x^p+y^q=y^r+z^p=z^q+x^r=2.
  • Page 74, second paragraph: “this can be true is of” should be “this can be true is if”
  • Page 76-77: The informal topological argument here does not quite work as stated, for if two rectangles with integer horizontal lengths (say) are connected by a common horizontal line segment rather than a common vertical one, then the lengths do not add together as suggested in the argument. To fix this, one needs a more complicated colouring scheme. Namely, one colours the interiors of rectangles with integer horizontal lengths green, and those with integer vertical lengths red. As for the edges, one colours the (open) vertical edges green and the (open) horizontal edges red. There are a few remaining corners which are not on any open edge that remain to be coloured; these can be assigned either red or green arbitrarily. With this colouring, any green path between the two horizontal edges of the big rectangle can be used to deduce the integer horizontal length of that rectangle, and similarly for a red path between the two vertical edges. (There are also several other proofs, for instance one can induct on the number of rectangles.)
  • In page 77, second-to-last paragraph, “it seems that the assertion is plausible” should be “it seems plausible that the procedure always terminates”.
  • In the diagrams on page 79 and page 82, the labels C and D should be switched.
  • In page 82, “X is a quarter-length or less from M” should be “X is a quarter-length or more from M”.
  • In page 87, second-to-last paragraph, “eliminated (c)” should be “eliminated is (c)”.
  • In page 90, second paragraph: “ths game” should be “this game”.
  • In page 92, third paragraph: “that it is a sure” should be “that is a sure”.  In the second to last line, “all sure losers, as we have shown above” should be “all sure winners, as we have shown above”.
  • In page 95: change all occurrences of “rouble” to “ruble” (for consistency).  In the third paragraph, “in terms of equation” should be “in terms of equations”.
  • In page 96, last line: “restricted to between” should be “restricted to lie between”.
  • In page 97, second paragraph: “s = \pm 4 (\hbox{mod }20)” should be “s = \pm 4 (\hbox{mod }20) or s = \pm 6 (\hbox{mod }20)“, and so the example values of s should now read s = 4, 6, 14, 16, 24, 26, 34, 36, \ldots.
  • In the references, “G.A. Hardy” should be “G.H. Hardy”.

— Errata to the Chinese version —

  • In p. 40(d), “如果k是正奇數，則 1^k+2^k+…+n^k 可被 n+1 整除”, n+1 should be n(n+1)/2.
  • In p. 76, “幸好第一個因數（p-1)i 和 p^2 是互質的（因為（p-1)i 和 p 是互質的）”, the i’s should be factorial symbols.
  • In p. 146, “與一些三角形和圓有關的∠APF，看起來似乎是比較「主流」的角”, APF should be ABF.
  • In p. 154, “若 n 為偶數，令 m=(n+2)/1”, (n+2)/1 should be n/2+1.
  • In p. 202, ” 我們只能說 s = ±4 (mod 20)，因此羊的數量可能是 4、16、24、36、44、56 等”, s = \pm 4 (\hbox{mod }20)” should be “s = \pm 4 (\hbox{mod }20) or s = \pm 6 (\hbox{mod }20)“, and so the example values of s should now read s = 4, 6, 14, 16, 24, 26, 34, 36, \ldots.

Thanks to Paulo Ventura Araújo, Thomas Drucker, Tathagata Gupta, Percival Li, ouzi, Cecil Rousseau, Naoki Sato, dsilvestre, Arnstein Skåra, Tom Verhoeff, Weiyu Xu, Li Yan, Au Cheuk Yin, and 현민 오 for corrections.