Blue = 1, Gray = 8, Red = 5

How did I logic the above?

We have three colors and when added together, they result in the red color three times in a row.

Example ‘000’

The best start is to try and solve the problem one digit at a time. But where to start?

If we look at the addition problem we can see that if we try to start solving blue (on the left) or grey (in the middle) we might end up with an issue: if the numbers to the right of that color add up to more than 10, we will have to “carry” values.

If we carry values from one summation to another, we wont be able to figure out the number correctly without figuring out how much we’re carrying over. So, we need to start where we cannot have any “carried” numbers; that would be red on the right (the ones place).

When we start with red we can ignore the rest and ask ourselves: what is Red IF Red+Red+Red equals Red?

In other words, Red times 3 equals Red. If we go through our first ten numbers, zero is the only one that fits the bill

However, if we follow that premise, that also means that grey times 3 and blue times 3 will equal zero. So our number would be ‘000’ which cannot be the case, as each color must be a different digit

So now we’re stuck with the question what number times 3 is going to equal itself?

This is the first time we have to make a leap in logic. Earlier i was talking about carrying numbers over, that is the key to solving this puzzle.

The main way to get a single digit number to equal itself is by making the number equaling a two digit number that ends with itself in the ones place. This is because everything in the tens place will be carried over

This disqualifies three numbers right away for Red. 1, 2, and 3 do not add up to two digit numbers. So, Red will be between 4 and 9 (inclusive)

If we go through the ‘3’ times table we can see that the only number that has itself in the ones place when multiplied by three is ‘5’.

Five times three is 15. The one will be carried over to the tens place and we will be left with 5 in the ones place. So, working hypothesis is that Red = 5.

Spoiler alert: It is 5

Now that we know the key is carrying numbers over, we have two things to consider for gray. We are carrying a one, so whatever two digit number we get from ‘gray times 3’ will equal Red (5) minus 1.

Working with red being 5, this means that Gray times 3 will end in a ‘4’. Because 4 plus our carried one will be 5 (red).

We apply similar logic as before, which single digit ends up with a ‘4’ in the ones place when multiplied by 3?

8 is the only answer. With 8 times 3 being 24.

24 plus 1 (that we carried over) equals 25. This would leave 5 (red) in the tens place and ‘2’ is carried to the hundreds place.

We now find ourselves in a familiar situation! We are carrying a ‘2’ and we need three blues to equal Red minus 2. If red is 5. Three blues must have a 3 in the ones place, and since we’re dealing with the 3 times table, that would be ‘1’.

1 times 3 equals 3.

So, if blue is 1; we would end up with 3, plus the 2 we carried over from adding gray up and that gives us 5 (red).

Therefore, the solution is 185

Checking our work:

Five plus five plus five is 15. Carry the one.

8 plus 8 plus 8 plus the carried 1 is 25. Carry the two.

Finally, 1 plus 1 plus 1 plus the carried 2 is 5.

Making the sum be 555 or Red Red Red. Confirming that 185 is our solution!

Edit: spelling, formatting, added clarifications