One board, a few numbers. Everyone races to find the one that is the sum of two others.
Greg Tang, an American author of math books for children, invented it. His first Kakooma Puzzles book came out in 2010. Scholastic published an Addition Edition and a Multiplication Edition in 2012.
You already know
Adding. 3 plus 5 is 8. You know it without thinking.
The strange part
But a board with 5 numbers has 10 different pairs to add. A board with 9 has 36. Add them all and someone else has already tapped.
You might ask
Is there a way to skip most of those additions?
Learn to play: just three rules
1
A board holds a few numbers. Exactly one of them is the sum of two others.
2
Everyone looks at once. The first right tap scores a point. A wrong tap locks you out of that round.
3
The first to reach the agreed number of right answers wins.
A mathematician answers
Kakooma looks like a race of quick eyes. It is really a race of smart choices. Know which numbers to skip, and you tap first.
Why do you say that?In the order you look at one round: count the additions, then compare sizes, then check odd and even.
Before you look
Adding every pair takes a lot of additions
Major premise
Every two numbers make a pair, and each pair is one addition. Each number pairs once with every other number.
Minor premise
With 5 numbers, each pairs 4 times: 5×4=20. But 3 with 5 and 5 with 3 are the same pair, counted twice.
Conclusion
20÷2=10 additions. A board of 9 needs 36. A round lasts only 20 or 30 seconds, so you have to choose.
First glance
The answer is a sum, so start from the largest
A sum is bigger than both numbers you add. So the two smallest numbers can never be the answer.
To test a number, subtract: take away another number on the board and see if the difference is there too.
In 8, 3, 12, 5, 1, try 12, then 8. 8−5=3. Found it, after only 5 subtractions.
Second glance
An odd number gives itself away
Major premise
Odd plus odd is even. Even plus even is even. Only odd plus even is odd.
Minor premise
For an odd number to be the answer, one of its two parts must be another odd number.
Conclusion
If the board has only one odd number, it is never the answer. Don’t even look at it.
Before you look How many pairs? Slide and count
10 lines, one line for each addition
Tap me
Drag me
Numbers5
Additions10
5 × 4 ÷ 2 = 10
Each number pairs with the other 4: 5×4=20. Every pair got counted twice, so divide by 2.
Line up the additions for every board. What do you notice?
6
10Tap me
15
21
28
36
6, 10, 15, 21, 28, 36. Each step adds exactly 4, 5, 6, 7, 8 more: the new number pairs with every old one. These are triangular numbers. Each one can be laid out as a triangle of dots.
First glance Start from the largest, test by subtracting
Tap me
Five numbers: 8, 3, 12, 5, 1. The answer is a sum, and a sum is bigger than its parts. So try the largest, 12, first.
0 subtractions
Second glance An odd number gives itself away
Tap me
Orange is odd, blue is even. 7 is the only odd number. An odd answer needs odd plus even, and there is no second odd number. Skip 7. Among the rest, 2+8=10.
odd + odd = even
even + even = even
odd + even = odd
Multiplying has a pattern too: only odd times odd is odd. One even number makes the product even.
You don’t add every pair,the one who chooses wins.
The moment you choose right,you see the beauty of math.