Puzzle for July 27, 2023 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Once again, we thank Abby S (age 12) for sending us an interesting puzzle! Thank you so much, Abby!
Scratchpad
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Hint #1
eq.6 may be written as: A - B = D + F - E In the above equation, replace D + F with C (from eq.4): eq.6a) A - B = C - E
Hint #2
Add the left and right sides of eq.6a to the left and right sides of eq.2, respectively: B + E - A + A - B = C - E + C - E which becomes E = 2×C - 2×E Add 2×E to both sides of the above equation: E + 2×E = 2×C - 2×E + 2×E which makes 3×E = 2×C Divide both sides by 2: 3×E ÷ 2 = 2×C ÷ 2 which makes eq.2a) 1½×E = C
Hint #3
In eq.3, replace C with 1½×E: 1½×E + D = B + E Subtract E from each side of the equation above: 1½×E + D - E = B + E - E which becomes eq.3a) ½×E + D = B
Hint #4
In eq.5, substitute ½×E + D for B (from eq.3a): ½×E + D = F - D Add D to both sides of the equation above: ½×E + D + D = F - D + D which becomes eq.5a) ½×E + 2×D = F
Hint #5
Substitute 1½×E for C, and ½×E + 2×D for F (from eq.5a) in eq.4: 1½×E = D + ½×E + 2×D which becomes 1½×E = 3×D + ½×E Subtract ½×E from each side of the above equation: 1½×E - ½×E = 3×D + ½×E - ½×E which makes E = 3×D
Hint #6
Substitute (3×D) for E in eq.2a: 1½×(3×D) = C which makes 4½×D = C
Hint #7
Substitute (3×D) for E in eq.3a: ½×(3×D) + D = B which becomes 1½×D + D = B which makes 2½×D = B
Hint #8
Substitute (3×D) for E in eq.5a: ½×(3×D) + 2×D = F which becomes 1½×D + 2×D = F which makes 3½×D = F
Hint #9
Substitute 2½×D for B, 4½×D for C, and 3×D for E in eq.6a: A - 2½×D = 4½×D - 3×D which becomes A - 2½×D = 1½×D Add 2½×D to both sides of the equation above: A - 2½×D + 2½×D = 1½×D + 2½×D which makes A = 4×D
Solution
Substitute 4×D for A, 2½×D for B, 4½×D for C, 3×D for E, and 3½×D for F in eq.1: 4×D + 2½×D + 4½×D + D + 3×D + 3½×D = 37 which simplifies to 18½×D = 37 Divide both sides of the above equation by 18½: 18½×D ÷ 18½ = 37 ÷ 18½ which means D = 2 making A = 4×D = 4 × 2 = 8 B = 2½×D = 2½ × 2 = 5 C = 4½×D = 4½ × 2 = 9 E = 3×D = 3 × 2 = 6 F = 3½×D = 3½ × 2 = 7 and ABCDEF = 859267