Class 9-10

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Some problems for the Senior School.

In this edition of ‘Adventures’ we study a few miscellaneous problems, some from the PRMO and some from the AIME (the ‘American Invitational Mathematics Examination’). As usual, we pose the problems first and present the solutions later.

We know that Pythagorean triples are infinite in number, and the most common formula for generating triples is to take two relatively prime odd numbers s and t, where s > t ≥ 1, and produce the triple (st, (s2−t2)/2 , (s2+t2)/2 ). However, can we generate all possible triples from just one triple? Can we generate infinitely many triples from just one triple? These might be questions worth investigating.

Graphics calculators have been available to students in secondary school in some countries now for more than thirty years, although of course their capabilities have been developed in various ways to support the school curriculum over that time. The most frequent use of these devices seems to be concerned with the representation of functions, including in particular their graphical representation, which was an important component of a previous paper in this magazine (Kissane, 2016).

There is another class of semi-regular 3D shapes - the Archimedian solids. In these the faces are again regular polygons but they are not all congruent, i.e. they differ in the number of sides. One such Archimedian solid is the truncated  icosahedron.

On the web page [1] belonging to A2Y Academy For Excellence, there appears a curious method to find the square root of a four-digit number if it is known that that number is a perfect square. We describe the method here using two examples and then consider how to explain it.

I must confess that it was the top layer that attracted me to quilting. This was clearly a case of the whole being greater than the sum of its parts- I was wonderstruck at how scraps of material could be pieced together to make beautiful patterns that were all at once eye-catching and pleasing.

A classroom observation. A colleague a shared an observation that had come up during an exploratory class he was taking at the middle school level. The topic being discussed was Pythagorean Triples, i.e., triples (a, b, c) of positive integers satisfying the relation a 2 + b 2 = c 2 . If the three integers also happen to be relatively prime to each other, i.e., share no common factors exceeding 1, then the triple is called a ‘Primitive Pythagorean Triple’ (PPT for short).

Magic triangles can be added to each other term by term, the same way that magic squares can be added to each other. We show here how two third order magic triangles can yield another third order magic triangle through addition.


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