## What Are Matching Riddles?

Matching riddles are another classic logic puzzle. They generally present information about some situation and ask you (typically through deduction and a process of elimination) to figure out what objects belong to whom, or something along those lines. A classic example is Einstein's Riddle.

## How Are Matching Riddles Valuable Mathematically?

While it is possible to solve a matching riddle through brute force by constructing a giant grid, it is much more instructive (and efficient) to make deductions about the puzzle through careful study. Making these deductions teaches students to find a point of attack on a complicated question, looking for the entry that allows the rest of the question to be understood and solved.

Additionally, students learn to methodically track their progress; to apply deductive logic; and to try multiple cases. By trying a case that fails, students get an introduction to proof by contradiction.

For example, you could follow up matching puzzles by doing problems like the one below that require students to work on steps carefully and keep track of them to find the answer.

Square $$ABCD$$ has sides of length $$12$$ units each. Points $$W$$, $$X$$, $$Y$$, and $$Z$$ lie on sides $$AB$$, $$BC$$, $$CD$$, and $$DA$$, respectively, so that $$AW = \frac{1}{2}AB$$, $$BX = \frac{1}{2}BC$$, $$CY = \frac{1}{3}CD$$, and $$AZ = \frac{1}{4}DA$$. What is the area of quadrilateral $$WXYZ$$? (See image at right. From MATHCOUNTS 2010, Chapter Target Round #2.)

Matching puzzles and similar ones also often appear on the Specialized High School Admissions Test, which many of these students will be taking one year later. It will serve them well to master these questions now.

## How Might This Class Run?

Ultimately, the class is yours to run in whatever way you feel will best serves the students. Here is one potential timeline.

• 3 days: Introduce basics of the riddles, and solve progressively more difficult riddles.
• 2-3 days: Move on to more challenging riddles, and introduce twists to them. For example, you might discuss the mathematical meaning of "if" and introduce riddles that use it, of the form, "If Mayra plays soccer, then Jossiel will not play soccer."
• 4-5 days: Solve math problems that use deductive reasoning, case work, working methodically, and the idea of finding a contradiction to disprove an idea. Draw explicit parallels between these math problems and the methods used to solve logic puzzles previously.

## Problems To Use

In addition to Einstein's Riddle, many of these puzzles can be found by Googling terms like "Einstein riddles" or "grid puzzles". (For example, here are several more.) Moreover, it would be beneficial to take similar puzzles from the Specialized High Schools Admissions Test so that students can acclimate to it.