Puzzle for February 6, 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
In eq.4, replace F with A + C + E (from eq.3): A + C + E – D = A + B + E Subtract A and E from both sides of the above equation: A + C + E – D – A – E = A + B + E – A – E which becomes eq.4a) C – D = B
Hint #2
In eq.2, replace B with C – D (from eq.4a): C – D + C = D which becomes 2×C – D = D Add D to both sides of the equation above: 2×C – D + D = D + D which becomes 2×C = 2×D Divide both sides by 2: 2×C ÷ 2 = 2×D ÷ 2 which makes C = D
Hint #3
In eq.4a, substitute C for D: C – C = B which means 0 = B
Hint #4
Subtract A from both sides of eq.5: C + D – A – A = A + E – A which becomes C + D – 2×A = E Substitute C for D in the above equation: C + C – 2×A = E which becomes eq.5a) 2×C – 2×A = E
Hint #5
Substitute B + C for D (from eq.2) in eq.6: F – A = A – B + C + B + C which becomes F – A = A + 2×C Add A to both sides of the above equation: F – A + A = A + 2×C + A which becomes eq.6a) F = 2×A + 2×C
Hint #6
Substitute 2×C – 2×A for E (from eq.5a), and 2×A + 2×C for F (from eq.6a) in eq.3: A + C + 2×C – 2×A = 2×A + 2×C which becomes 3×C – A = 2×A + 2×C In the above equation, add A to each side, and subtract 2×C from both sides: 3×C – A + A – 2×C = 2×A + 2×C + A – 2×C which makes C = 3×A and also makes D = C = 3×A
Hint #7
Substitute (3×A) for C in eq.6a: F = 2×A + 2×(3×A) which means F = 2×A + 6×A which makes F = 8×A
Hint #8
Substitute (3×A) for C in eq.5a: 2×(3×A) – 2×A = E which becomes 6×A – 2×A = E which makes 4×A = E
Solution
Substitute 0 for B, 3×A for C and D, 4×A for E, and 8×A for F into eq.1: A + 0 + 3×A + 3×A + 4×A + 8×A = 19 which simplifies to 19×A = 19 Divide both sides of the equation above by 19: 19×A ÷ 19 = 19 ÷ 19 which means A = 1 making C = D = 3×A = 3 × 1 = 3 E = 4×A = 4 × 1 = 4 F = 8×A = 8 × 1 = 8 and ABCDEF = 103348