Sharif Quansah and Hayden Leroux won prizes for partial answers.
The problem: Calculate the remainder when 23 is divided by 3
when 25 is divided by 5
when 27 is divided by 7
when 35 is divided by 5
when 37 is divided by 7
when 45 is divided by 5 Continue reading
This is the crispest proof I’ve found that the complex numbers ℂ are the only finite field extension of the real numbers, ℝ. Continue reading
How far apart should the centres of the two circles be for the three areas A, B, and C to be equal? Continue reading
The work-energy principle we study in M2 is closely related to the principle of least action used in more advanced physics. Continue reading
No prize-winners this time. Though I know (because they told me) that the year 13 Further Maths students worked out answers, none of them wrote down the answers and handed them in, so no prizes.
If all the people of the world stood shoulder-to-shoulder (with the babies in slings, and the old and sick people held up by the people next to them), how much area would we cover? Continue reading
The problem: prove that the crescent-moon shaped region, upper left, has the same area as the shaded triangle.
Mariama Bah, Shannon Bradley, and Sharif Quansah all handed in correct solutions and won prizes.
Proof: Let AO be 1 unit. Then the area of the triangle is ½ unit, and AB is √2 units, by Pythagoras.
The area of the semicircle ABC is 2π
The area of the quarter-circle AFBO is π
The area of the segment AFBD is π−½
The area of the semicircle AEB is π
The area of the crescent-shape = area of semicircle AEB minus area of segment AFBD
=area of triangle ∎
This investigation is interesting in itself and uses almost all the ideas we learn in S2. Continue reading