# ABCD - Math Puzzle

by Andrew
(New York)

The four numbers a < b < c < d can be paired in six different ways. If each pair has a different sum, and if the four smallest sums are 1, 2, 3, and 4,
what are all possible values of d?

### Comments for ABCD - Math Puzzle

 Nov 01, 2011 ABCD - Math Puzzle by: Staff Question: by Andrew (New York) The four numbers a < b < c < d can be paired in six different ways. If each pair has a different sum, and if the four smallest sums are 1, 2, 3, and 4, what are all possible values of d? Answer: Order is not important No repetition of the same combination Permutations possible = n!/[r!(n-r)!] n = 4 r = 2 Permutations possible = 4!/[2!(4-2)!] = 4!/[2!(4-2)!] = (4*3*2*1)/(2*1*2*1) = 6 {a,b} {a,c} {a,d} {b,c} {b,d} {c,d} The six possibilities for the sum the pairs: a + b a + c a + d b + c b + d c + d since a < b < c < d, the combinations with the four smallest sums are: a + b a + c a + d b + c Therefore, the four smallest sums are: a + b = 1 a + c = 2 a + d = 3 b + c = 4 This is a linear system of four equations Solve for c b + c = 4 b - b + c = 4 - b 0 + c = 4 - b c = 4 - b a + c = 2 solve for a substitute 4 - b for c a + (4 - b) = 2 a + 4 - b = 2 a + 4 - 4 - b = 2 - 4 a + 0 - b = -2 a - b = -2 add the following two equations: a + b = 1 a - b = -2 -------------- a + a + b - b = 1 - 2 2a + 0 = -1 2a = -1 divide each side of the equation by 2 2a/2 = -½ a*(2/2) = -½ a*(1) = -½ a = -½ substitute -½ for a in the following equation a + d = 3 -½ + d = 3 -½ + ½ + d = 3 + ½ 0 + d = 3 + ½ d = 3 + ½ d = 3.5 the final answer is: d = 3.5 Thanks for writing. Staff www.solving-math-problems.com

 Nov 02, 2011 only answer? by: Anonymous is 3.5 the only answer?

 Nov 02, 2011 Math Puzzle by: Staff 3.5 is the only answer because it is the only value of d which will satisfy the linear system of four equations: a + b = 1 a + c = 2 a + d = 3 b + c = 4 although the numerical solutions for all four letters were not included with the answer (all the question asked for was the value of d), they are: (a=-1/2, b=3/2, c=5/2, d=7/2) Thanks for writing. Staff www.solving-math-problems.com

 Oct 25, 2017 Only Answer? NEW by: Anonymous Pretending that we dont know what a,b,c,d are yet, how can you presume that a+d>b+c? Try solving the problem, but this time do b+c>a+d

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