Puzzle for December 19, 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.
Scratchpad
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Hint #1
Multiply both sides of eq.5 by (–1): (B – E) × (–1) = (D – A) × (–1) which becomes E – B = A – D In eq.3, replace E – B with A – D: A – D = B – A – D Add D and A to both sides of the above equation: A – D + D + A = B – A – D + D + A which makes 2×A = B
Hint #2
In eq.3, replace B with 2×A: E – 2×A = 2×A – A – D Add 2×A and D to each side of the above equation: E – 2×A + 2×A + D = 2×A – A – D + 2×A + D which becomes E + D = 3×A which may be written as eq.3a) D + E = 3×A
Hint #3
Add E and F to both sides of eq.4: D – F + E + F = A – E + F + E + F which becomes D + E = A + 2×F In the above equation, substitute 3×A for D + E (from eq.3a): 3×A = A + 2×F Subtract A from both sides: 3×A – A = A + 2×F – A which makes 2×A = 2×F Divide both sides by 2: 2×A ÷ 2 = 2×F ÷ 2 which makes A = F
Hint #4
Substitute A for F, and 2×A for B in eq.2: C + A = 2×A Subtract A from both sides of the above equation: C + A – A = 2×A – A which makes C = A
Hint #5
Substitute A for C and F, 2×A for B, and 3×A for D + E (from eq.3a) in eq.1: A + 2×A + A + 3×A + A = 32 which means 8×A = 32 Divide both sides of the equation above by 8: 8×A ÷ 8 = 32 ÷ 8 which makes A = 4 making C = F = A = 4 B = 2×A = 2×4 = 8
Hint #6
Substitute 4 for A in eq.3a: D + E = 3×4 which means eq.3b) D + E = 12
Hint #7
In eq.6, subtract D from both sides, and add E to each side: E – D – D + E = A + B + D – E – D + E which simplifies to 2×E – 2×D = A + B Substitute 4 for A, and 8 for B in the equation above: 2×E – 2×D = 4 + 8 which becomes eq.6a) 2×E – 2×D = 12
Hint #8
Substitute D + E for 12 (from eq.3b) in eq.6a: 2×E – 2×D = D + E In the equation above, add 2×D to both sides, and subtract E from both sides: 2×E – 2×D + 2×D – E = D + E + 2×D – E which makes E = 3×D
Solution
Substitute 3×D for E in eq.3b: D + 3×D = 12 which makes 4×D = 12 Divide both sides of the above equation by 4: 4×D ÷ 4 = 12 ÷ 4 which means D = 3 making E = 3×D = 3×3 = 9 and ABCDEF = 484394