Logic Puzzle # 78 Logic Problems Help

Logic Problem Solution:
Christmas Tree, O Christmas Tree

Neither the blue spruce (clue 5), Douglas fir (6), Fraser fir (7), nor Scotch pine (8) was the fifth and last tree bought; the white pine was. By clue 3, the Starrs bought their tree immediately before the family who chose the Douglas fir did; while by clue 7, the family who bought the Fraser fir picked their tree immediately ahead of the Lights. The only possible commonality between the two clues is if the Starrs bought the Fraser fir and the Lights chose the Douglas fir. However, by clue 3, then, the Lights' Douglas fir would have cost \$25 more than the Starrs' Fraser fir, but by clue 7, the Starrs' Fraser fir would have cost \$50 more than the Lights' Douglas fir. So, four of the five tree-buying families are named between clues 3 and 7. Since the Ball family didn't pick the Fraser fir (2) and bought their tree later in the evening than the family choosing the Douglas fir (6), the Balls are the fifth family to the four in clues 3 and 7. There are three possible orderings for the five families given clues 3, 6, and 7: 1) the Starrs, the Douglas fir, the Balls, the Fraser fir, and the Lights; 2) the Starrs, the Douglas fir, the Fraser fir, the Lights, and the Ball family; or 3) the Fraser fir, the Lights, the Starrs, the Douglas fir, and the Ball family. In the first arrangement, the Lights would have bought the white pine, contradicting clue 4. Trying the second arrangement, the Ball family would have gotten the white pine. The Starrs would have picked the Scotch pine (8) and the Lights then the blue spruce. Since the Scotch pine cost \$30 less than the Garlands' tree (8), the Garlands wouldn't have bought the Douglas fir, which cost \$25 more than the Starrs' Christmas tree (3). The Garlands would have bought the Fraser fir and the Hollys the Douglas fir. Letting the blue spruce equal X in price, the Fraser fir would have cost X + 50 (7), the Hollys' Douglas fir 2X (5), and the Starrs' Scotch pine 2X - 25 (3). The Starrs' Scotch pine also would have cost X + 20 (8), so that 2X - 25 = X + 20. Solving, X would equal \$45. So the Lights would have spent \$45, the Garlands \$95, the Hollys \$90, the Starrs \$65, and the Balls \$60 for their white pine (4). However, there would be two differences between tree prices of \$5--impossible by clue 1. Therefore, arrangement 2) also fails, and arrangement 3) remains: the family selecting the Fraser fir bought first, followed in order by the Lights, the Starrs, the family who chose the Douglas fir, and the Ball family, who paid \$60 for the white pine they picked (4). The Garlands didn't buy the first tree (8); they picked the Douglas fir, and the Hollys chose the Fraser fir. Since the Garlands' Douglas fir cost \$25 more than the tree the Starrs bought (3), the Starrs didn't take home the Scotch pine, which cost \$30 less than the Garlands' choice (8). The Starrs' Christmas tree is the blue spruce and the Lights' the Scotch pine. Letting the Starrs' blue spruce cost X, the Garlands' Douglas fir cost X + 25 (3). The Hollys' Fraser fir cost 2X (5), so the Lights' Scotch pine was 2X - 50 (7). The Lights' Scotch pine also cost X - 5 (8). So, 2X - 50 = X - 5. Solving, X equals \$45. So, the Starrs spent \$45, the Garlands \$70, the Hollys \$90, and the Lights \$40. The five families bought Christmas trees last evening as follows:

• 1st - Holly family, Fraser fir, \$90
• 2nd - Light family, Scotch pine, \$40
• 3rd - Starr family, blue spruce \$45
• 4th - Garland family, Douglas fir, \$70
• 5th - Ball family, white pine, \$60

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