Cross Products
This multiplication puzzle is either 3 by 3 involving each of the numbers 1 to 6 exactly once, or 4 by 4 involving the numbers 1 to 8 exactly once.
The challenge
Fill in some of the squares, two numbers for each row and each column, so that the product of the numbers in a row is the number marked to the far left and the product of the numbers in a column is the number marked above the column. Some rows or columns may not be marked – if so, there is no constraint on the product of those rows or columns.
Solving examples
Solve the puzzle above by finding columns and rows where you can identify the two numbers. The 30 column must have 5 and 6, and the 10 row must have 2 and 5. Next, the 12 column must have 3 and 4 and the 4 row must have 1 and 4. The rest follows quickly.
The 5 row in this second puzzle must have 1 and 5, the 21 row has 3 and 7, the 32 row has 4 and 8, and the top row has, by elimination, 2 and 6. The 7 column has 1 and 7, the 40 column has 5 and 8, the 18 column has 3 and 6, and the second column has 2 and 4. Put this together for the solution.
How to create
Make these by filling in the numbers on the inside of the puzzle first, writing down the products, and then removing all the interior numbers.
Helping your child
Puzzles are meant to be challenging and to take time, so please don’t ruin the fun by telling your child how to do them. These puzzles are chosen so that you can create them easily and then have fun solving them together.
If your child gets stuck on a puzzle, you have several options. You can, of course, give very small hints, if you can think of things that won’t give away the puzzle. You can suggest looking at smaller or simpler versions of the puzzle. Encourage your child to be bold in their ideas, even if sometimes they lead to dead ends. We all learn a lot from our mistakes and dead ends! Let your child know that it is perfectly okay not to solve a puzzle on the first (or second or third) try, and that useful ideas may occur to them if they leave the puzzle alone for a day or two.
These puzzles are meant to be fun and to teach problem solving. One of the greatest mathematical pleasures is that AHA moment, after many false starts and much wrestling with a problem, when the answer is finally discovered – be sure to let your child experience that feeling of discovery as many times as you can!