Black and white tokens
There are 8 tokens which are colored either black or white noted with A , B , C , D , E , F , G and H. Below are a number of statements about their colors and your task is to be completely sure about their color based on these statements since you don't know which color is which :
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Neither B nor C has the color of D.
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G is white if and only if E has the same color as H.
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A and G have different colors if and only if E and F have the same color.
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G is white if and only if D is white.
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D is black excepting the case in which E has the same color as F.
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A doesn't have the same color as B if either A , either G is white.
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A , F and G don't have the same color.
Write the answer as the concatenation of the values 0 and 1 where 0 means black and 1 means white in the order A to H.
For example if tokens B C D and F are black and the other white the answer would be 10001011.
The answer is 1100101.
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1 solution
Firstly it would be useful to note that all statements are speaking of the colors of some tokens depending on how the colors of the other tokens are and in order to understand better how all the tokens should be colored such that the entire number of statements remains true therefore in order to find the configurations which are consistent with the entire set of statements it might be useful to express all that the statements say at a simpler expression. In order to do this consider firstly all the statements which speak of D this being statements 1 , 4 and 5 which cover therefore all information for how the color of D is in relation with the colors of other tokens from what it is already given from which what is found out can be formulated as the fact that D and G will have the same color which is different from the color of B and C (this being what can be concluded from statements 5 and 4 about the relation between the colors of D with others) and that that color is dependent on the state of colors of the pair of tokens E , F (this being derived from statement 6). Observe further that the state of colors of E and F does determine also the state of colors of A and G meaning that it speaks on whether the color of A is different from the color of G or not from statement 3. In this way it can be concluded that for assigning one value for (E,F) that is whether E and F have the same color that will determine independently the relation between A and G on one side and the color of D and G on the other side which because assigns a color to the pair D, G will speak of the relation between the color of A and G also. In the same way observe that if any of A and G will be white then A will have a different color from B (this being said from statement 6). So to see what was found out until now E , F determines the color of D , G and A and because determines that color determines the relation between A and B. This expression doesn't cover all the tokens (since token H is not taken into account until now) but speaks of the way most of the tokens are interdependent and affect each other based on the color of E and F and therefore can be said that is enough to check the consistency of the relation until now.
For the case in which it is assigned that E , F have the same color then A will have a different color from G and as there are just 2 possible colors would mean that one of A and G will be white which as a direct consequence by what is said in statement 6 (since "either A or G is white" is true) implies that A has a different color from B. This as A will have a different color from D and B thing which will imply that B and D have the same color leads to a contradiction as statement 1 would not be true. If E , F will have different colors then A , G , D will be black and one of E or F will be also black leading therefore until now to a consistent case remaining to check the other statements to complete everything. As from statement 7 A , G and F have different colors (meaning that not all have the same color) and A and G already have the same color it means F has a different color and therefore that F is white and E is black and since by statement 2 because G is not white E will have a different color from H meaning H is white. So it has been concluded that A , D , E , G are black and B , C , F , H are white and the set is consistent. This means that the answer is 01100101.