[OlymLSec_20160118PPPC] A Square Proof by Contradiction

Question

Explanation

     If  a + b = 11,  then  2ab = (a + b)² – (a² + b²) = 121 – 100 = 21.  But  2ab  is an even number, whereas  21  is odd.  This is a contradiction.  So (B) is impossible.  ©

Remarks

     Short and sweet isn’t it?  This uses the square-of-sum identity   (a + b)² = a² + 2ab + b².  I used the tactic of assuming the answer is correct  [H08]  and showing that this leads to something nonsensical [H05].  So the original assumption must be wrong.  This is called “proof by contradiction” or reductio ad absurdum (in Latin).

     By the way, the correct answer option is (E) from the Pythagorean Triplet   8² + 6² = 10²  with  {a, b} = {8, 6}.  The question seems to be taken from some Kangaroo mathematics competition.

Heuristics Used

H05. Work backwards

H08. Make suppositions

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* Lower Secondary Mathematics competition

* GCE ‘O’ Level “Elementary” Mathematics (challenge)

* other syllabuses that involve whole numbers and Pythagorean triplets

* any precocious or independent learner who loves a challenge

[OlymLS_20151220INEQ] An Egyptian Fraction Partition of Unity?

Problem

Introduction

     This question appeared in a “holiday homework” from an IP school.  It is not the usual type of question in exams and tests, but it is a good mental-stretching exercise.  It is asking us to split unity (the number “1”) into three unit fractions (a.k.a. Egyptian fractions).

     The first part can be solved by trial and error or “guess and check”.  The second part, proving that this is the only solution (are there any other solutions?), seems challenging.  We can solve this by making suppositions and using inequalities.

Solution

                                              

Heuristics Used

H03. Make a systematic list

H05. Work backwards

H07. Use guess and check

H08. Make suppositions  [ which may lead to negative conclusions ]

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* Primary School / Lower Secondary Olympiad

* other syllabuses that involve fractions and inequalities

* any learner who is itching for a mental challenge

[Pri20150303SJM] Race to the Bottom

Question

Thinking / Planning

Noting that Sarah spent all her money in both scenarios and that 22´(18´something) = 18´(22´something) , I am going to first guess that Sarah takes 18 days and 22 days to spend all her money in the 1^st scenario and the 2^nd scenario respectively. [ Heuristics: H07. Use guess and check & H08. Make suppositions ]  Then I try to adjust my educated guess.

Johari spent more money in the second scenario.  The difference in his spending among the two cases is 6, so we need

                10´(number of days₂) – 12´(number of days₁) = 6

Solution

If Sarah takes 18 days and 22 days respectively to spend all her money, then

the difference in Johari’s spending among the two cases is

                10´(22) – 12´(18) = 4

´³/₂:         10´(33) – 12´(27) = 6

Ans (a): Sarah has $22´27 = $594.
Ans (b): Johary has $(21+12´27) = $345.

Commentary

If the amounts of money involved were in the trillions, we would have thought that Sarah and Johari are certain countries, wouldn't we?

Anyway, a parent from a Facebook parent-support group posed this question asking for a simple solution without using ratios.  This question seems to be the equivalent of a system of simultaneous equations in four variables.  A few of us tried various methods to solve it, but all were quite complicated.  I knew this can be solved using algebra, but struggled for some time to give a simple solution.

I guess all solutions (even the one above) would have some notion of “ratio” hidden in it.  The reason is: every time you multiply or divide by something, there is actually a ratio involved.  But I hope this solution is “easy” enough.