Puzzle for December 19, 2020 ( )
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
Add C to both sides of eq.3: D – C + C = B + F + C which becomes D = B + F + C In eq.4, replace D with (B + F + C): E – F = A + B – (B + F + C) which is equivalent to E – F = A + B – B – F – C which becomes E – F = A – F – C Add F and C to each side of the equation above: E – F + F + C = A – F – C + F + C which becomes eq.4a) E + C = A
Hint #2
eq.1 may be written as: A + E + C + B + D + F = 24 In the above equation, replace E + C with A (from eq.4a), and B + D + F with A (from eq.2): A + A + A = 24 which makes 3×A = 24 Divide both sides of the above equation by 3: 3×A ÷ 3 = 24 ÷ 3 which makes A = 8
Hint #3
In eq.4a, substitute 8 for A: E + C = 8 Subtract E from both sides of the above equation: E + C – E = 8 – E which becomes eq.4b) C = 8 – E
Hint #4
Substitute B + D + F for A (from eq.2) in eq.5: B + D + F + D = B + E + F Subtract B and F from both sides of the equation above: B + D + F + D – B – F = B + E + F – B – F which simplifies to eq.5a) 2×D = E
Hint #5
Substitute 2×D for E in eq.4b: C = 8 – 2×D Substitute (8 – 2×D) for C in eq.3: D – (8 – 2×D) = B + F which is the same as D – 8 + 2×D = B + F which becomes eq.4c) 3×D – 8 = B + F
Hint #6
eq.2 may be written as: D + B + F = A Substitute 3×D – 8 for B + F (from eq.4c), and 8 for A in the above equation: D + 3×D – 8 = 8 which becomes 4×D – 8 = 8 Add 8 to both sides: 4×D – 8 + 8 = 8 + 8 which becomes 4×D = 16 Divide both sides by 4: 4×D ÷ 4 = 16 ÷ 4 which makes D = 4
Hint #7
Substitute 4 for D in eq.5a: 2×4 = E which makes 8 = E
Hint #8
Substitute 8 for E in eq.4b: C = 8 – 8 which means C = 0
Hint #9
Substitute 4 for D, and 8 for A and E in eq.6: B × 4 = 8 – 4 + 8 which makes B × 4 = 12 Divide both sides of the equation above by 4: B × 4 ÷ 4 = 12 ÷ 4 which makes B = 3
Solution
Substitute 4 for D, 0 for C, and 3 for B in eq.3: 4 – 0 = 3 + F Subtract 3 from both sides of the equation above: 4 – 0 – 3 = 3 + F – 3 which means 1 = F making ABCDEF = 830481