Ontario Mathematics Olympiad

Sample Question #3

The Missing Number

Many ancient civilizations have contributed to our mathematical knowledge. In Babylonia, Egypt, China, and India, as well as in the Greek, Roman, and Islamic Empires, the study of mathematics was highly valued, and all these civilizations made significant contributions to various areas of mathematics, including geometry, arithmetic, and algebra.

Many mathematicians have worked out interesting puzzles involving computations. Here is an example for you to work on:

Write down any two numbers that use all the digits from 0 to 9 exactly once. For example, you could write:

Next, add the two numbers. Finally, erase the original two numbers and any digit in the answer.

They understood that you could glance at the number 34 ? 282 and immediately say, "The missing number is 8".

Can you supply the missing digit in the following, and explain how it is possible to guess the missing digit in the answer so quickly?

Try to find the missing digit in the following numbers: