The Riddle of the Tiled Hearth is one of many mathematical puzzles from Henry Ernest Dudeney's 1907 book titled The Canterbury Puzzles And Other Curious Problems. The first group of puzzles in this book are based on the characters from Geoffrey Chaucer's Canterbury Tales. Puzzles from this book could be used as part of a cross curricular unit on history, literature and mathematics. There are a number of very interesting puzzles and games including the first pentomino puzzle called The Broken Chessboard and a clever variation of the game Nim called The Thirty One Game. Since this book was first published in 1907, the copyright has expired and it is freely available on Project Gutenberg.
It seems that it was Friar Andrew who first managed to "rede the riddle of the Tiled Hearth." Yet it was a simple enough little puzzle. The square hearth, where they burnt their Yule logs and round which they had such merry carousings, was floored with sixteen large ornamental tiles. When these became cracked and burnt with the heat of the great fire, it was decided to put down new tiles, which had to be selected from four different patterns (the Cross, the Fleur-de-lys, the Lion, and the Star); but plain tiles were also available. The Abbot proposed that they should be laid as shown in our sketch, without any plain tiles at all; but Brother Richard broke in,--
To use this activity with students, I would start by introducing using the tiled hearth story as written above. Then I would introduce some manipulatives that would let them physically explore and work with the puzzle. I would give each group of students a large 4x4 grid on a sheet of paper and some multi-link cubes of 4 different colours.
One aspect of this puzzle that I like is that students can play around with it and have some intermediate success. They might just place a few cubes on the grid. With time, they can refine their solutions to get better and better. Below shows how a student might explore to place more and more cubes.
The Solution from Canterbury Puzzles shows that the best solution leaves 3 blank spaces. Dudeney states, "The correct answer is shown in the illustration on page 196. No tile is in line (either horizontally, vertically, or diagonally) with another tile of the same design, and only three plain tiles are used. If after placing the four lions you fall into the error of placing four other tiles of another pattern, instead of only three, you will be left with four places that must be occupied by plain tiles. The secret consists in placing four of one kind and only three of each of the others." Below are both my solution using cubes and Dudeney's equivalent solution.
The final chapter of the Math at Work 12 textbook deals with Trigonometry and the Law of Sines and Law of Cosines. Towards the end of the chapter there is a puzzle (p351) that asks students to create a triangle using 9 of the numbers from 1 to 10. Each side of the triangle is the sum of 4 of these numbers. I liked the construct of this puzzle but I wasn't a big fan of the questions that it asked students so I decided to give it an overhaul. An image from the textbook is below.
We followed up this warm-up with an open middle style problem using the same situation that would require students to apply the law of cosines. A challenge like the one below gives the students a reason to practice the law of cosines without feeling tedious or repetitive.
Directions: Use the numbers 1-9 (using each number no more than once) to fill in the circles. The sum of the numbers on each side of the triangle is equal to the length of that side. What is the triangle with the largest (or smallest) angle that you can make?
A triangle with the largest angle (there are several variations with the same angle):
Side A: 6+8+9+7=30 Side B: 7+4+1+3=15 Side C: 3+2+5+6=16
Angle A: 150.799 Angle B: 14.119 Angle C = 15.082
A triangle with the smallest angle (there are several variations with the same angle):
Side A: 1+2+3+4=10 Side B: 4+8+9+7=28 Side C: 1+5+6+7=19
Angle A: 10.844 Angle B: 148.212 Angle C = 20.944
Another challenging question that could be asked is how many different arrangements of the numbers 1 to 9 in the triangle diagram could you make? You have to consider that rotations of the triangle are the same. This would be a challenging combinatorics question even for Pre-calculus 12 students.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 11 - G03 Solve problems that involve the cosine law and the sine law, including the ambiguous case.
Math at Work 12 - G01 Students will be expected to solve problems by using the sine law and cosine law, excluding the ambiguous case.
Math at Work 12 - N01 Students will be expected to analyze puzzles and games that involve logical reasoning, using problem-solving strategies.
Mathematics 12 - LR01 Analyze puzzles and games that involve numerical and logical reasoning, using problem-solving strategies
Pre-calculus 12 - PC03 Determine the number of combinations of n different elements taken r at a time to solve problems.