How many cars does the millionaire have?

Question

I had a logical riddle on a programming interview test which was something like this:

In his garage a millionaire had cars, of which only 2 were not white, only 2 were not green and only 2 were not red. How many cars did the millionaire have in his garage?

This seemed quite straightforward to me and I gave an answer, however I wasn't allowed to see my test afterwards and I cannot tell if the answer is correct.

I am curious to see if my answer was correct and also if there is a better mathematical derivation to the problem.

Answer 1

It could be

two cars (both black),

or

three cars, as AeJey explained.

Answer 2

A total of

three

cars, with colors distributed such that

one is red, one is green, and one is white.

If you take white, then all cars except two (green and red) are white. Likewise, if you take red, all cars except two (white and green) are red. And if you take green, all cars except two (red and white) are green.

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Reasoning for this conclusion

If there are more, then the statement that "of them two are not white/green/red" will fail.

Let the total number of cars be $n$
Total number of white cars be $x$
Total number of green cars be $y$
Total number of red cars be $z$

Then $$2 = n-x \rightarrow x = n-2$$ $$2 = n-y \rightarrow y = n-2$$ $$2 = n-z \rightarrow z = n-2$$

So $$n = x+y+z$$ $$n = (n-2) + (n-2) + (n-2) $$ $$n = 3n - 6$$ $$6 = 2n $$

Thus,

n = 3

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Note

As far as what I understood from the question "of which only 2 were not white" means there is at least one white car and definitely more than 2 cars. The same for all colors specified in the question. Otherwise the questioner should have mentioned "of which 2 were not white" or "2 cars were not white" or just "none of them are white" instead.

Answer 3

2 or 3.

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For a total of $n$ cars, of which $w$ are white, $r$ are red, $g$ are green, and $c$ have another colour:

$$\begin{align} n & , w, r, g, c \in \mathbb{Z}_{\ge 0} \tag{1} \label{eq1} \\ n & = w + r + g + c \tag{2} \label{eq2} \\ n & \ge c \tag{3} \label{eq3} \\ 2 & = n - w \implies w = n - 2 \tag{4a} \label{eq4a} \\ 2 & = n - r \implies r = n - 2 \tag{4b} \label{eq4b} \\ 2 & = n - g \implies g = n - 2 \tag{4c} \label{eq4c} \\ n & = (n-2) + (n-2) + (n-2) + c && \text{From $\eqref{eq2}, \eqref{eq4a}, \eqref{eq4b}, \eqref{eq4c}$}\\ n & = 3n - 6 + c \\ 2n & = 6 - c \\ n & = 3 - \tfrac{c}{2} \tag{5} \label{eq5} \\ c & = 0 \land n = 3 \Large \lor \normalsize c = n = 2 && \text{From $\eqref{eq1}, \eqref{eq3}, \eqref{eq5}$}\\ \end{align}$$

As we see, when taking $\eqref{eq1}$ and $\eqref{eq3}$ into account, $\eqref{eq5}$ has two solutions, at $c=0$ and $c=2$.
If $c$ were any bigger, we'd have $c>n$ meaning that our millionaire has more other coloured cars than the total number of cars, which would be quite impossible.

• If $c=0$ then $n=3 \implies w=r=g=n-2=1$
• If $c=2$ then $n=2 \implies w=r=g=n-2=0$

So either he has one white, one red, and one green car, or he has two cars that are neither white nor red nor green.

The first solution corresponds to AeJey's answer, the second one to FlorianF's answer.

Answer 4

Two possible answers hinging on the answer to a question; are red, green and white the only possible colors the cars could be? This meta-question is likely the real test of the puzzle; being able to spot the "hole" in the requirements given by the puzzle (that not all cars in the real world are either white, red or green).

If the puzzle enforces an alternate reality where these three colors are the only options even for millionaires, then the answer is

three cars; one white, one red, one green.

In this case, for each of the three colors, only

one

of the cars can have that color, and therefore two cars cannot.

If we're modelling the real real world, then the answer is

two

cars, all in a color other than white, red, or green, such as black, silver or blue. In this situation, the same cars are not white, not red and not green, because they're black (or blue or silver or yellow or...).

Answer 5

Given the ambiguities in the question (does it mean just the minimum possible, does it mean that they have to be red, white, green, define red/white/green), I think the only really correct answer is:

Possibly 2 or 3

For what it's worth, I agree with the idea that the test is looking for someone seeing the issues with the spec.

Answer 6

Let $w$, $g$, $r$ be the numbers of white green and red cars, and $x$ be the number of cars of any other color

$w + g + r + x = total$

$total - w = 2 = g + r + x$
$total - g = 2 = w + r + x$
$total - r = 2 = w + g + x$

$w + g + x = g + r + x$
$w = r$

$g + r + x = w + r + x$
$g = r$

therefore

$w = r = g$

but

$x <= 2 - w - g$
$x <= 2 -2r$
$x <= 2 (1-r)$

$r >= 0$
$x >= 0$
$1-r >= 0$

by the above constraints,
$r <= 1$ AND $r >= 0$

$r = g = w = 1$ and $x = 0$ or $r = g = w = 0$ and $x = 2$

therefore

1 green, one white and one red
or
2 of any other color, and none green white or red

Answer 7

All the answers already given either (1) don't explain why the answer is what it is or (2) try to do so by setting up equations and doing a pile of algebra. Here's another approach. First of all, I assume every car is of exactly one colour (which may or may not have been a deliberate ambiguity in the original question).

First of all, there are at least two cars (since there are e.g. two that aren't red). If there are exactly two cars then two (hence both) the cars are not red, two (hence both) the cars and not green, and two (hence both) the cars are not white. So if there are exactly two cars then they are of colours other than red, green and white, and clearly in this case the given conditions hold.

Otherwise

there are at least three cars. At most two of them are not red, hence at least one is red; ditto for green and white. So we have at least one car of each of these colours, and these of course provide two that aren't red, two that aren't green, and two that aren't white. So there can't be any more cars (if we do, and e.g. it isn't white, then we have too many cars that aren't white). Hence, if there are more than two cars then there are exactly three, one of each of the specified colours.

Personally I find this clearer than all the algebra, though SQB's calculation does have its own appeal.

Answer 8

I propose that this millionaire might have

arbitrarily many cars.

Among these cars

all but two of them are white with red and green polka dots,

while the other

two cars are both blue.