Puzzle for November 1, 2019 ( )
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.
* BC and DE are 2-digit numbers (not B×C or D×E).
Scratchpad
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Hint #1
Subtract B from both sides of eq.4: E + F – B = B + D – B which becomes E + F – B = D In eq.3, replace D with E + F – B: A – B = E + F – B – E which becomes A – B = F – B Add B to both sides of the equation above: A – B + B = F – B + B which makes A = F
Hint #2
In eq.2, replace F with A: D + A = A + C Subtract A from each side of the equation above: D + A – A = A + C – A which means D = C
Hint #3
In eq.5, substitute A for F, and C for D: C + C + E = B + A which may be written as eq.5a) A + B = 2×C + E In eq.3, substitute C for D: eq.3a) A – B = C – E
Hint #4
Add the left and right sides of eq.3a to the left and right sides eq.5a, respectively: A + B + (A – B) = 2×C + E + (C – E) which simplifies to 2×A = 3×C Divide both sides of the above equation by 2: 2×A ÷ 2 = 3×C ÷ 2 which makes A = 1½×C and which also makes eq.5b) F = A = 1½×C
Hint #5
Substitute 1½×C for A in eq.5a: 1½×C + B = 2×C + E Subtract 1½×C from both sides of the equation above: 1½×C + B – 1½×C = 2×C + E – 1½×C which becomes eq.5c) B = ½×C + E
Hint #6
eq.6 may be written as: 10×B + C + D = 10×D + E Substitute C for D in the equation above: 10×B + C + C = 10×C + E which becomes 10×B + 2×C = 10×C + E Subtract 2×C from both sides: 10×B + 2×C – 2×C = 10×C + E – 2×C which becomes eq.6a) 10×B = 8×C + E
Hint #7
Substitute (½×C + E) for B (from eq.5c) in eq.6a: 10×(½×C + E) = 8×C + E which becomes 5×C + 10×E = 8×C + E Subtract 5×C and E from both sides of the above equation: 5×C + 10×E – 5×C – E = 8×C + E – 5×C – E which means 9×E = 3×C Divide both sides by 3: 9×E ÷ 3 = 3×C ÷ 3 which makes 3×E = C and which also makes D = C = 3×E
Hint #8
Substitute (3×E) for C in eq.5b: F = A = 1½×(3×E) which means F = A = 4½×E
Hint #9
Substitute (3×E) for C in eq.5c: B = ½×(3×E) + E which means B = 1½×E + E which makes B = 2½×E
Solution
Substitute 4½×E for A and F, 2½×E for B, and 3×E for C and D in eq.1: 4½×E + 2½×E + 3×E + 3×E + E + 4½×E = 37 which simplifies to 18½×E = 37 Divide both sides of the equation above by 18½: 18½×E ÷ 18½ = 37 ÷ 18½ which means E = 2 making A = F = 4½×E = 4½ × 2 = 9 B = 2½×E = 2½ × 2 = 5 C = D = 3×E = 3 × 2 = 6 and ABCDEF = 956629