Puzzles and Riddles

Learning Goals:

  • Multiply and divide whole numbers using a variety of tools and strategies
  • Determine the missing numbers in equations
  • Develop and apply reasoning skills to make and test math predictions

The Challenge:

The goal of these three-by-three yohaku puzzles is to find which boxes to place the numbers 1-9, so that each row and column multiplies to the corresponding products at the right and bottom. You can repeat digits, if you want!

Here is an example that has been solved. You can see in the centre column that 4 x 2 = 8, and then 8 x 9 = 72.

An example solution of a three-by-three yohaku puzzle where numbers need to multiply down to the products 40, 72, and 30 (at the bottom), as well as across to the products 48, 50, and 36 (down the right side.) From left to right, the solved digits are 2, 4, 6 (top row), 5, 2, 5 (middle row), and 4, 9, 1 (bottom row).

A beginner difficulty three-by-three yohaku puzzle where numbers need to multiply down to the products 4, 18, and 70 (at the bottom), as well as across to the products 24, 42, and 5 (down the right side.) The centre left, bottom left, and bottom middle digits have all been filled in as 1's.

A medium difficulty three-by-three yohaku puzzle where numbers need to multiply down to the products 24, 72, and 210 (at the bottom), as well as across to the products 24, 120, and 126 (down the right side.) The top right digit has been filled in as a 4, and the bottom middle digit has been filled in as a 9.

A medium difficulty three-by-three yohaku puzzle where numbers need to multiply down to the products 40, 48, and a missing number (at the bottom), as well as across to the products 28, 144, and 90 (down the right side.)

A three-by-three yohaku puzzle where numbers need to multiply down to the products 60, 21, and 288 (at the bottom), as well as across to the products 24, 40, and 378 (down the right side.)

Questions and Prompts to Support your Child:

  • What have you tried so far? What could we try together?
  • If that combination of numbers didn’t work out, what could you try instead?
  • If you feel stuck, it might help to start by thinking about the products around the outside, listing potential sets of three numbers that could multiply to get these answers. Then start looking for matching numbers from these lists – those could be the numbers that work in the intersecting blanks!

Extensions & Adaptations:

  • If calculations are feeling too tricky, grab a calculator!
  • This might be our first time experiencing multiplication focused around more than 2 factors. Try thinking about pairing up factors in different ways before multiplying with the third. How could we represent multiplication with three factors, since 2-dimensional arrays will no longer quite make sense?
  • Do any of these puzzles have more than one possible solution? How can you prove it?
  • Try creating your own yohaku puzzle by starting with nine different digits in the boxes, multiplying to get the products to the right and below, then erasing the original numbers
  • Check out yohaku.ca for many more puzzles – including some more challenging options!

Adapted from Yohaku.ca and NRICH Math