## Extended Thinking Series: Column Addition

Development of our interactive exercises is a very exciting process. It results not only in what you find at HappyNumbers.com, but also in so many interesting things discussed and discovered by the team along the way. In our Advanced Exercises Series we share exercises with non-standard solutions which we’ve stumbled upon while doing our work. You may find some of them useful for challenging the advanced students in your class (or the whole class at some point!). Enjoy and subscribe to our blog for more Extended Thinking Series ideas!

For this post we decided to talk about the exercises on column addition. Here is one posted in our Twitter some time ago:

Solving non-standard exercises is not about remembering the method or trying all possible combinations; it’s about logic and finding the “weak” places in the exercise (though it may take a lot of time and many attempts)!

In this particular case, what one should notice is that if you add two 3-digit numbers you can get 4-digit numbers. But only those which start with 1, because the largest 3-digit number is 999, so the largest possible sum of two 3-digit numbers is 1998. This is the weak place of the exercise, so you start with: A is equal to 1 and can be nothing else!

Finding A gives us a lot of information:

What’s next? Let’s see what kind of other weak places we may find. Let’s explore the ones column: 1+B=C doesn’t give us much of clue. But in the tens and hundreds columns there is something: in the hundreds column we have 1+B=11. That might be possible if B=0, but that doesn’t work here. However, if B=9 and we have a carry over of 1 from the tens column, then this solution works, because you have 1+9 in the tens column!

And now it’s easy to see that C=0!

## More ideas

You may simplify the exercise, by making it a 2-digit number + 2-digit number = 3-digit number, something like AA + BB = AAC. You may do the opposite too, by using larger numbers like AAAAA + BBBBB = AAAAAC. Either way, it’s still the same logic to solve it.

## Even more ideas

Do you think this is still too easy for your students? Here’s what you can do to make it more challenging:

1. Shuffle the letters, for example:

2. Still too easy? Try switching to subtraction, for example:

That’s it! That’s our Advance Exercise for today.

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