Noticing the clues
Piecing together, 甲 throws out a hint about 乙 and 丙 having different numbers while knowing their combined sum is 14.
Assessing 甲's insight
甲 realizes 乙 and 丙 must have different numbers, which means 甲's number doesn't make (14 - A)/2 an integer, preventing simultaneous equal values for 乙 and 丙.
Evaluating even distributions
甲's number, if even, would make 乙 and 丙 potentially equal. Since 甲 knows they are different, his number must be odd or result in odd numbers.
Examining the implications
OK, let me see. If 甲 sees a 2, 乙 and 丙 being equal could be plausible. However, this would make 甲's claim impossible, indicating 甲 did not see a 2.
Evaluating implications
For 甲's statement to hold true, 甲 cannot see an even number card since it could allow for identical 乙 and 丙 values. The sum of the other cards should be odd.
Breaking down possibilities
Interestingly enough, for 甲's statement to be valid, 甲 shouldn't have seen an even number. This conclusion aligns with 甲's position, deeming certain even numbers inconsistent.
Pinpointing the issue
Noticing a problem with 甲 having a 13: the sum of the other two cards would be 1, which isn't possible since standard playing cards only go up to 13.
Charting out options
With 甲 ruling out even numbers and 13, there are six valid possibilities for 甲's card: 1, 3, 5, 7, 9, and 11.
Taking a closer look
乙, seeing his own card, confirms that 甲,乙, and 丙 each have unique cards, ensuring no duplicates exist among them. This deduction relies on the distinctiveness of each card value.
Ensuring card uniqueness
乙’s confidence in each card being unique implies there are no valid scenarios where any two cards share the same value while summing to 14.
Ensuring unique combinations
乙’s certainty that all cards are distinct means no valid scenarios include duplicate cards summing to 14, leading to the conclusion that no such duplicates exist.
Evaluating unique values
By checking each possible B value, I’m ensuring no scenarios exist where 甲, 乙, and 丙 share the same value while summing to 14. This confirms distinct values.
Ensuring distinct sums
A closer look at B=7 shows no duplicates in 甲 and 丙's cards. This confirms all cards are unique, aligning with 乙's certainty.
Laying out the options
Interestingly enough, with B=9, all three cards stay distinct, confirming 乙's view. Checking B=8 and B=10 shows possible duplicates, so they don't fit 乙's position.
Mapping out combinations
OK, let me see. Considering B=7, B=9, and B=11, there are valid combinations of 甲 and 丙's cards that ensure all three remain distinct. B=13 isn't an option.
Evaluating combinations
Now I'll map out valid combinations for 甲, 乙, and 丙's cards that sum to 14, considering the constraints and ensuring all are unique.
Identifying valid triples
First, let's list valid triples (A, B, C) that sum to 14, ensuring all cards are distinct and match the statements given. Each combination must be systematically examined.
Validating combinations
I’m mapping out valid triples (A, B, C) that sum to 14, ensuring all cards are distinct. Invalid combinations are excluded, and distinct pairs are carefully examined for each value of B.
Ensuring validity
Piecing together the valid triples that sum to 14, I confirm they align with 甲's and 乙's statements. Now, I'll check if 丙 can deduce the numbers based on his own card.
Weighing C values
Evaluating unique C values that pinpoint a single valid A and B configuration. This shows if 丙 can identify which numbers 甲 and 乙 have at once.
Evaluating potential outcomes
Analyzing whether 甲's and 乙's perspectives about distinct values fit. Addressing C=4 as a possible clue for combining valid A and B sets while ensuring all values remain unique.
Navigating ambiguity
Interestingly, C=4 doesn't clarify the distribution for 丙, leaving two valid combinations. However, C=2 narrows it down to three combinations, which still leaves room for uncertainty.
Mapping out the options
I determined that the only unique solution for C is when C=6, pigeonholing the values for A and B as (1,7,6). This eliminates all ambiguity.
Thought about card numbers and logic,持续 51 秒