Arithmetic, hold the boredom: digit puzzles

One of the big problems with teaching arithmetic is that we carefully structure the learning process to be boring. Drill is effective, but only if the students do it, and some will not. In this post we present a challenging series of puzzles (and techniques for creating more puzzles) to wash some of the boredom out of learning arithmetic. The picture below shows the solution to a viral puzzle called four fours. The puzzle asks the student to make the numbers 0 to 20 out of four copies of the digit four. The term digit is used judiciously because it’s not quite the number four. Notice that “44” counts as two fours. The number of other symbols used is unlimited and sum, difference, product, quotient, power, square root, factorial, and parenthesis are all available. The solutions shown are far from unique. Consider 4+4+4+4=16 or 4×(4/4+4)=20, for example. This post extends the problem and provides keys for five fives and six sixes as well as proposing additional activities.

Solving the four-fours puzzle is an activity for fourth grade or higher. Notice that the solution for 19 uses the factorial, and in fact Occupy Math needed the factorial in all three puzzle keys in this post. The need for the factorial symbol may cause this activity to be an enrichment activity — and an opportunity to teach the factorial. If you are going to use a calculator, make sure it has a factorial button or supply a table of values: 0!=1, 1!=1, 2!=2, 3!=6, 4!=24, 5!=120 and 6!=720 should be those that are needed.

Four is a perfect square — which helps a lot.

The fact that the square root of 4 is 2 opens up additional options for the four fours puzzle. On the other hand, having five digits or six digits to work with causes a combinatorial explosion in the number of formulas that can be written, which creates a different reason that the five fives and six sixes puzzle might be easier. Occupy Math does supply keys for the five fives and six sixes puzzle.

Notice the solution for 20 in the five fives puzzle does use square roots, showing a way they can be useful even though the square root of five is an irrational number. The fact that any number to the power zero is one shows up as a way to burn several digits to make a one. The trick of putting two digits to make 44, 55, and 66 as part of solutions is quite helpful. Simply solving each of the three puzzles is a good activity, but there is more that can be done with these puzzles.

Occupy Math also has a key for the seven sevens puzzle. If you would like a copy, actually a neat PDF with the three keys in this post and the seven sevens, please send a request to Occupy Math at dashlock@uoguelph.ca.

Structuring the activities for these puzzles

Solving each of these puzzles can be structured as competitive group work. Put the numbers 0-20 on the board. Teams send people up to fill in solutions for each number, awarding a point to the first team to find a way to generate each number. It may be worth putting hints on the board reminding students of square roots, factorials, and what a zero power does if the teams are having trouble solving some problems. The idea of using 44, 55, and 66 might also need to be suggested by the instructor, if no-one thinks of it in the first ten minutes.

The puzzle can be extended by allowing multiple different solutions. Many of the numbers have several possible formulas that can generate them. While you are free to come up with your own scoring system, Occupy Math suggests awarding three points for the first solution, two for the second, and one for each additional formula found.

Why stop at 20?

The short answer to this question is that you have to stop somewhere and 20 is big but not too big. If you want to use this with younger children, you might stop at ten or twelve, which lets you avoid factorials. The other reason for this is that, at some point, you will encounter a number that cannot be constructed: 20 is small enough. The reason Occupy Math knows that all the numbers from 0-20 can be done with five fives and six sixes is that he worked out and typed up the keys. Of course not stopping at 20 also creates some activities.

• See how much farther you can go: 21, 22, … This can be a long-term activity. Make a poster and add things to it as class members discover them.
• Seven sevens? Eight eights? Occupy Math does not know if eight eights will work, but this is a good question for clever students.
• Another possibility is finding the simplest formula for a given number.
• Some numbers can be expressed in many, many different ways; pick a number and try to find as many different ways to compute it as you can. This leads to another type of contest: award the point for a number to the team that finds the most ways to express the number. If there is a tie, both teams get the point.

What does this activity teach?

The big payoff for this activity is practice with arithmetic operations. It can be structured to teach operator precedence as well. The obvious payoff is that it stretches the mental muscles and gives practice with problem solving. If students work in teams, it is also a good exercise for learning the skills of cooperative problem solving. Having cooperating teams compete with one another can give practice in cooperation while still using the spark supplied by competition.

If students are putting solutions on the board, a good thing to do is to praise innovative solutions in a manner that encourages other students to adopt those innovations in subsequent work. This might include things like discovering the availability of 44 by using two fours. A variation on this activity might be for the students to draw four or five digits, possibly different from one another, from a hat and then use those to generate as many numbers as they can. The question “how many numbers from zero to one-thousand can you generate?” suggests a very long term activity. There are actually many possible additional puzzles that can be generated in the spirit of the ones presented here. Occupy Math would like to thank Rachel Brown for pointing out the four fours puzzle that led to this post.

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics