Puzzle for March 24, 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.2, replace A + B with D + E (from eq.5): C + E = D + E – C In the equation above, subtract E from each side, and add C to both sides: C + E – E + C = D + E – C – E + C which makes 2×C = D
Hint #2
In eq.5, replace B with A + D (from eq.3): D + E = A + A + D which becomes D + E = 2×A + D Subtract D from each side of the above equation: D + E – D = 2×A + D – D which makes eq.3a) E = 2×A
Hint #3
Add C to both sides of eq.6: B – C + C = C + D + C which becomes B = 2×C + D In eq.3, substitute 2×C + D for B: 2×C + D = A + D Subtract D from both sides of the equation above: 2×C + D – D = A + D – D which makes 2×C = A
Hint #4
Substitute (2×C) for A in eq.3a: E = 2×(2×C) which makes E = 4×C
Hint #5
Substitute 2×C for A, and 2×C for D in eq.3: B = 2×C + 2×C which makes B = 4×C
Hint #6
Substitute 2×C for D, and 2×C for A in eq.4: C – 2×C = 2×C – F which becomes –C = 2×C – F Add C and F to both sides of the above equation: –C + C + F = 2×C – F + C + F which makes F = 3×C
Solution
Substitute 2×C for A and D, 4×C for B and E, and 3×C for F in eq.1: 2×C + 4×C + C + 2×C + 4×C + 3×C = 32 which simplifies to 16×C = 32 Divide both sides of the equation above by 16: 16×C ÷ 16 = 32 ÷ 16 which means C = 2 making A = D = 2×C = 2 × 2 = 4 B = E = 4×C = 4 × 2 = 8 F = 3×C = 3 × 2 = 6 and ABCDEF = 482486