# Number Triple Puzzles This week on Occupy Math, we have a new puzzle to help make arithmetic practice go down more smoothly. Look at the array of numbers above. To play the puzzle you take groups of three numbers, either in a row or in an corner-shaped group of three squares. Examples appear below the fold. The numbers in the example above are chosen so that the groups of three numbers, when added, include all the numbers from zero to twenty seven. The goal? Gotta find them all. Warning: if you write down a 4×4 array of digits at random, then your odds of being able to get all the numbers are bad. Occupy Math used a design trick to create that example.

The picture below shows some of the moves you can make while solving the puzzle. There are five moves shown and they capture the following numbers.

• Orange: 0+0+0=0,
• Green: 2+1+0=3,
• Brown: 3+3+0=6,
• Violet: 3+4+7=14,
• Blue: 9+9+9=27. Notice that the smallest possible total is 0 and the largest is 27, at least if we are adding three digits. The example array of numbers is carefully designed to have all the numbers from zero to twenty-seven possible. A sheet of twenty of these puzzles – including the example above – is available here. Notice there there are two ways to have three squares in a row, and there are four ways to put down an angle. These are displayed below. When working the puzzle, the shapes used to capture numbers may overlap, or, to put it another way, you may use any one of the digits as many times as you like. Here are four puzzles, from the PDF linked above, in case you are in a big hurry to have some puzzles. All four of these puzzles let the students find all twenty eight possible numbers, from zero to twenty seven. Who is this puzzle for?

This puzzle is intended to encourage arithmetic practice, requiring students to add three digits. It probably works for second through sixth grade as a problem to be worked individually or in small groups cooperating to find all the numbers. The competitive version, described in the next paragraph, is good for fourth through ninth grade. If you fill in the numbers randomly, the resulting puzzle usually has many of the possible numbers, but not all of them. In particular notice the three zeros and three nines placed to permit the numbers zero and twenty seven to be possible.

This puzzle can also be used as a timed, competitive exercise. The puzzle is put on the board by the teacher and then students, or small teams, have a time limit during which they try find as many numbers as they can. For second graders, ten minutes would be a good limit. Eighth or ninth graders should be allowed two minutes, and of course the teacher is free to adjust the timing as they wish. Another interesting timing method, for the brave instructor, is to work the puzzle yourself and then call time when you finish — this permits the students to try their mettle against their teacher.

This puzzle is also intended for parents that either want extra math exercises for their students or who need more practice exercises. If you need more puzzles, feel free to mail Occupy Math at dashlock@uoguelph.ca. The puzzle generator can make thousands of these puzzles in a few minutes.

A comment on the size of the puzzle: why 4×4?.

If we had a 3×3 board, then there would be only 22 groups of three squares, of the sorts allowed. This makes getting the numbers from zero to twenty seven impossible! A 4×4 board has 52 groups of three squares and so getting all the possible sums of three digits is possible, but a little tricky. A 5×5 board has 94 scoring configuration and even random boards have a good chance of hitting all the numbers from zero to twenty-seven, as long as the three nines and three zeros are sitting in scoring configuration.

There are a couple of enrichment exercises you can draw from looking at how this puzzle works. The first is to have students design their own puzzle, trying to get all the possible sums to appear. This is a little difficult but far from impossible. The other is to work out the counting formula for the number of scoring configurations (three squares in a row and three forming an angle) that can be put down on a board with side length N. The answer is:

2×N×(N-2)+4×(N-1)2
Be sure to ask the student for an explaination. A good hint is to look at the square with side length 3 and think about how that answer worked.

Bonus scores

The boards supplied by Occupy Math, above, are all carefully crafted, using digital evolution, to have solutions that include all the possible numbers. It can be interesting to add bonus scores that emphasize other mathematical principles. The score for one of these puzzles is just the number of different numbers captured. To this basic score you might add:

1. Five points for the student(s) that have the longest sequence of adjacent numbers, e.g. 2,3,4,5,6,7,8,9.
2. Five bonus points for the student(s) that have the most prime numbers. The possible primes that can show up in these puzzles are 2, 3, 5, 7, 11, 13, 17, 19, and 23
3. Three points to the student(s) with the most even number.
4. Four points to the student(s) with the most multiples of three.
5. Five points to the students with the most Fibonacci numbers. The possible Fibonacci numbers in the puzzle’s range are 1,2,3,5,8,13, and 21
6. Five points to the student with the most unique numbers — numbers no one else found.

Harder version of the puzzle.

If we let students multiply the three numbers, as well as add them, then they can find a lot more numbers, and puzzles larger than 4×4 become more useful. The bonus categories still apply — except that no prime numbers can be achieved by multiplying two or more numbers bigger than one. The most unique numbers bonus should be worth twenty points if you allow multiplication. Here are four 5×5 puzzles. If you allow addition and multiplication then there are 94 numbers that can be found in the upper two puzzles and 93 in the lower two. The digit zero was available to the design software,which was maximizing the number of distinct numbers in the array, but even one zero wipes out several possible triples under multiplication (it makes them all zero) so the puzzle design software just avoided zero.

The use of the two ways to join three squares for the scoring positions in these puzzles was deliberate. One or two squares is really easy and, when you hit scoring shapes with four squares, the number of possible scoring positions becomes huge and this makes the puzzle, perhaps, less fun. If you have really sharp students that need an additional challenge, then adding some 4-square scoring positions might be a good idea. As always, remember that this activity post joins the list of activity posts. Also please remember that comments and suggestions are welcome by e-mail (dashlock@uoguelph.ca) and in the comments section.

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