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A Knight's Journey
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
8 Methods for Three by One
This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?
Age 16 to 18
Challenge Level
Use the scalar product of the diagonals. As the quadrilateral is flexed the diagonals change but the lengths of the sides are constant. All the vectors change but the the squares of the vectors of the sides (representing the lengths) remain constant. To preserve symmetry and obtain this scalar product in the form required, write
$$2{\bf d}_1={\bf a}_1-{\bf a}_2-{\bf a}_3+{\bf a}_4, \quad 2{\bf d}_2=-{\bf a}_1-{\bf a}_2+{\bf a}_3+{\bf a}_4.$$
NRICH is part of the family of activities in the Millennium Mathematics Project.