Recreational Mathematics

Puzzles involving shapes and figures

Another group of puzzles include dissecting and rearranging a shape to form another shape. Take for example this cross made up of 5 squares of equal size. (2) How do you dissect it with 2 straight lines to form a square? (Hint: use tessellation). Other such puzzles can be solved by sawing or stripping.

Another type of dissecting puzzle does not involve rearranging. E.g. (3) what is the maximum number of regions that can be obtained with 7 straight line dissecting a circle?

Paper-folding is also an entertaining mathematical game. Origami is the art of paper-folding, which allows us to fold a plane sheet of paper into a three-dimensional object, like a crane, a jackstone, etc. The proof that origami is a mathematical art can be seen in the fact that there is even a function in Mathematica for origami! We'll not go into origami per se, but will take a look at simpler paper-folding games. (4) What is the largest equilateral triangle that can be folded from a plane sheet of square paper? No markings/measurements allow on the paper.

Graph Theory also has its place in mathematical recreation. Try drawing the figures below without lifting your pen from the paper or retracing any line.
Figure (5a)	Figure (5b)

(6) Draw 4 straight lines through the 9 dots without lifting your pen from the paper.

(7) Matchstick puzzles involve moving/adding/removing matches to achieve certain configurations. For the figure on the right, (a) Move 2 matches to make 6 squares; (b) Move 4 matches to make 10 squares; (c) Remove 2 matches to make 2 squares.

Dominoes can be considered as 2 squares joined together to form a rectangle. The number on each square is from the set {0,1,2,3,4,5,6}and are represented by the number of dots rather than the numerals themselves. There are 28 distinct permutations. Consider a subset of this set of 28 consisting of the 6 dominoes 0-0, 0-1, 1-1, 0-2, 1-2, 2-2. (8) How do you arrange these 6 dominoes to form a rectangle which adds up to the same number of dots on each side of the rectangle?

Because dominoes come in 2 colours, it is sometimes mixed with a chessboard to form a bigger set of puzzle. The typical chessboard is an 8X8 checkered board. (9)(a) If the 2 white corner squares are removed, is it possible to tile the board with dominoes? (b) What about if the squares marked A and B are removed?

(10) Consider a constrained chessboard with only 9 squares. How can the black knights and white knight exchange position using the rules of chess, i.e. knights move in an L-shape?

Suppose the dominoes are extended to form trominoes (3 squares), how many possible tromino configurations are there? There are 2. Rotation is allowed. How about tetrominoes (4 squares)? There are 6. 5 if flipping is allowed. (11) How many pentominoes (5 squares), assuming both rotation and flipping are allowed? Those who have played the game of Tetris will be familiar with these shapes. The game involves such shapes dropping down from the top and the player has to determine where to place the shapes so as to clear these shapes at the bottom. The way these shapes are cleared depends on the rules: it could be 3 or more squares forming a straight line; these squares may have to be of the same colour; diagonals may be allowed.

Rubik's Cube was very popular game in the 1980s. It's a cube made up of 27 smaller cubes of the same size and has 6 different coloured faces. (12) How do you get from the plain cube to the configuration on the right?

Now we move from puzzles involving shapes and figures to word puzzles. Next Page.

Answers:

(1) It is impossible to rearrange the 15 Puzzle so that 14 and 15 are interchanged. This is due to the theorem that only arrangements that have an even number of interchanges has a solution.

(2)

This one can be solved by tessellation.

(3) 29. The answer follows the formula n(n+1)/2 + 1, where n is the number of straight lines. How is this formula obtained (Hint: every new line added must cut all previous lines) and is it still valid for large values of n?

(4) Fold the square in half to make crease EF. Fold AB so that B lies on EF. This gives points G and H. Fold HGJ. While B is on G, fold AB back to AH to get K. AJK is the largest equilateral triangle (prove this).

(5) Part (a) has a solution, but there is no solution to (b). (b) is the famous Konigsberg Bridge Problem (see Euler). There is only solution to such problem if there are at most 2 vertices with odd number of lines and one of the odd-numbered vertex will be the start, while the other will be the end (if there are 2 of them).

(6)

(7)

                     (a)                                              (b)                                                (c)

(8)

(9) (a) It is impossible because there are more black squares than white ones. Reason: each domino spans across 1 black and 1 white square. (b) Yes. As long as A and B does not cut the spiral into any section with odd number of squares.

(10) It can be achieved after 4 clockwise (or anti-clockwise) moves for each knight.

(11) 12.

(12) Apply 180^o turns to the following faces: front, right, back, front, right, back.