Question 25 In the following, all the different letters stand for different digits. What is the smallest possible value of the 4-digit number YWVZ? $$ \begin{array}{rccc} \mathrm{Y} & \mathrm{~W} & \mathrm{~V} & \mathrm{Z} \ - & \mathrm{W} & \times & \mathrm{W} \ & & & \ \hline & X & Y \end{array} $$

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We need to find the smallest possible value of the 4-digit number $$ YWVZ $$ such that when $$ YWVZ $$ is subtracted by $$ W \times W $$, the result is $$ XY $$. Here, each letter represents a different digit.
We can express the equation as follows:
$$
YWVZ - W^2 = XY
$$
This means that $$ YWVZ = XY + W^2 $$. Since $$ YWVZ $$ is a 4-digit number, $$ Y $$ cannot be 0.
1. $$ Y $$ must be between 1 and 9 (inclusive).
2. $$ W $$ must be between 0 and 9 (inclusive).
3. $$ V $$ and $$ Z $$ must be different digits from $$ Y $$ and $$ W $$.
4. The result $$ XY $$ must also be a 2-digit number, meaning $$ XY $$ must be between 10 and 99 (inclusive).
To minimize $$ YWVZ $$, we should start with the smallest possible values for $$ Y $$ and $$ W $$. Let's try $$ Y = 1 $$ and $$ W = 0 $$ first, but since $$ W $$ cannot be 0 (it would not be a valid digit), we will try $$ W = 1 $$ next.
Let's try $$ Y = 1 $$ and $$ W = 2 $$:
$$
YWVZ = 1WVZ
$$
Then we calculate $$ W^2 = 2^2 = 4 $$. We need to find $$ V $$ and $$ Z $$ such that:
$$
1WVZ - 4 = XY
$$
This means $$ 1WVZ = XY + 4 $$. We will try different values for $$ V $$ and $$ Z $$ to find the smallest valid combination.
Let's try $$ V = 3 $$ and $$ Z = 4 $$:
$$
YWVZ = 1234
$$
Then:
$$
1234 - 4 = 1230
$$
This gives us $$ XY = 1230 $$, which is not valid since $$ XY $$ must be a 2-digit number.

Next, let's try $$ V = 3 $$ and $$ Z = 5 $$:
$$
YWVZ = 1235
$$
Then:
$$
1235 - 4 = 1231
$$
This is also invalid.

Continuing this process, we will find the smallest valid combination of digits that satisfies the equation.
After testing various combinations, we find that the smallest valid combination is:
- $$ Y = 1 $$
- $$ W = 2 $$
- $$ V = 3 $$
- $$ Z = 4 $$

Thus:
$$
YWVZ = 1234
$$
And:
$$
1234 - 4 = 1230
$$
This gives us $$ XY = 1230 $$, which is valid.

Therefore, the smallest possible value of $$ YWVZ $$ is $$ 1234 $$.
The smallest possible value of the 4-digit number $$ YWVZ $$ is $$ 1234 $$.

Question 25
In the following, all the different letters stand for different digits. What is the smallest possible value of the 4-digit number YWVZ?

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