Lately, we are seeing more questions of this nature.

In the above picture, each chess symbol represents a digit from 0 to 9. Different symbols represent different digits. What are the values of each symbol – **♙**,** ♘** and **♖**?

### Elementary Math Concepts

This is one of our favorite type of questions. Not only does it involve understanding of the standard algorithm, place values and regrouping, it also requires the student to apply number sense and intuition. To solve it, the student may also apply some guess-and-check work.

### So What are the Values?

First we note that two single digit numbers **♙**+**♙** is enough to push a double digit number **♘♘** to a third place value. This tells us that a) **♘♘** is a rather large 2-digit number and b) **♖♖♖** is a rather small 3-digit number.

Also, we know the largest value **♙**+**♙** can be is 18 (=9+9), and the largest **♘♘** is 99. So, that maximum value **♙**+**♙**+**♘♘** can be is 117 (=18+99). However **♖♖♖** has to be made up of the same three digits, so **♖** must be 1.

So, let’s try **♖♖♖** = 111 and **♘♘** = 99. See if we can find a whole single-digit number **♙** that works.

**♙**+**♙** + 99 = 111

Yes. **♙** = 6 would work. So, **♙** = 6, **♘** = 9 and **♖** = 1.

#### What about other combinations?

Since **♖♖♖** is fixed, let’s try **♘♘** = 88, then **♙** would have to be (111 – 88)/2 = 11.5 which is neither a single digit number nor a whole number. By induction, we see that any other smaller values of **♘♘** would not work too, because **♙** would have to be larger than 11.5.

#### The answer

So, the only plausible answer is **A = 6, ♘ = 9 and ♖ = 1**.

Kim DoanPlease help me solve this question. Thank you so much!

How many different pairs (a, b) can be formed using

numbers from the list of integers {1, 2, 3, …, 10} such

that a < b and a + b is even?