# Kakooma Sums, Products, Fractions and Create Your Own Kakooma

The rules for Kakooma are simple. Greg Tang developed this puzzle; I first heard him speak as a parent of elementary school children in 2002. Mr. Tang had a part in inspiring me to change my career to teach math. Subsequently, I attended one of his teach the teacher workshops.

In each nine-number square (or 4-#, or 6-#, 7-#) , find the one number that is the sum of two other numbers. Use all nine sums to create one final “puzzle-in-a-puzzle” and solve.

This can be done with multiplication and the numbers can be fractions.

The most challenging aspect of the lesson was exploring how difficult it is to make a Kakooma puzzle. I had the children choose two addends (a number which is added to another number) such as 6 and 7 with a solution of 13. I had them create a small grid on the flip side of the page and record 13 as the solution in the corresponding square. Then I challenged them to fill in the other 6 squares with numbers that would not be obviously eliminated, such as 100. One student said, “let’s use 1.” But another students said, “we can’t use 1 since 1 + 6 = 7, so that would give us two solutions.” So we recorded the number 2. Now, 4, 5, and 11 are eliminated from being the next number we use because they are the differences between 6 and 2, 7 and 2, and 13 and 2. We also have to eliminate the sums of 8, 9, and 15 since adding 2 to our existing numbers would create those multiple solutions. The students quickly realized how difficult it was to create a puzzle square with only one solution. Another reason to have a lot of respect for Mr. Tang.

Multiplication puzzles are seemingly easier since the answers often appear as the largest number in the set. Creating a challenging multiplication puzzle would require having the solution not be the largest number and have products that are close together, such as 42, 45, 49, 54, 56, 63, and so on (6x7, 9x5, 7x7, 9x6, 8x7,9x7).

The fraction puzzles I did with my advanced class was very challenging as it required creating equivalent fractions with common denominators. For example, if most of the numbers had a denominator of 27 and there was a fraction 8/9, first convert that fraction to 24/27 and then find the sum solution.

I will welcome students next week to share their Kakooma creations with me so I can solve them. Beware of multiple solutions and focus on the end puzzle.

Have fun!!

Attachment | Size |
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Kakooma_Sums_Create_Your_Own_K-1.pdf | 2.24 MB |

Kakooma_Sums_Create_Your_Own_2-3.pdf | 1.69 MB |

Kakooma_Sums_Products_Create_Your_Own_Puzzle_3-5.pdf | 2.64 MB |

Kakooma_Fraction_Sums_5-6.pdf | 641.68 KB |