Puzzle for April 14, 2022 ( )
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 D and E to both sides of eq.4: E – D + D + E = B – E + D + E which becomes eq.4a) 2×E = B + D Add F and D to both sides of eq.3: D – F + F + D = B – D + F + D which becomes eq.3a) 2×D = B + F
Hint #2
Add the left and right sides of eq.4a to the left and right sides of eq.3a, respectively: 2×D + 2×E = B + F + B + D which becomes eq.4b) 2×(D + E) = 2×B + F + D
Hint #3
In eq.4b, replace D + E with B + C + F (from eq.6): 2×(B + C + F) = 2×B + F + D which becomes 2×B + 2×C + 2×F = 2×B + F + D Subtract 2×B and F from each side of the equation above: 2×B + 2×C + 2×F – 2×B – F = 2×B + F + D – 2×B – F which becomes eq.4c) 2×C + F = D
Hint #4
In eq.5, substitute 2×C + F for D (from eq.4c): F – C = 2×C + F – F which becomes F – C = 2×C Add C to both sides of the equation above: F – C + C = 2×C + C which makes F = 3×C
Hint #5
Substitute 3×C for F in eq.4c: 2×C + 3×C = D which makes 5×C = D
Hint #6
Substitute (5×C) for D, and 3×C for F in eq.3a: 2×(5×C) = B + 3×C which becomes 10×C = B + 3×C Subtract 3×C from each side of the above equation: 10×C – 3×C = B + 3×C – 3×C which makes 7×C = B
Hint #7
Substitute 7×C for B, and 5×C for D in eq.4a: 2×E = 7×C + 5×C which becomes 2×E = 12×C Divide both sides of the above equation by 2: 2×E ÷ 2 = 12×C ÷ 2 which makes E = 6×C
Hint #8
Substitute 7×C for B, and 6×C for E in eq.2: 7×C – 6×C = A – 7×C which becomes C = A – 7×C Add 7×C to both sides of the equation above: C + 7×C = A – 7×C + 7×C which makes 8×C = A
Solution
Substitute 8×C for A, 7×C for B, 5×C for D, 6×C for E, and 3×C for F in eq.1: 8×C + 7×C + C + 5×C + 6×C + 3×C = 30 which simplifies to 30×C = 30 Divide both sides of the above equation by 30: 30×C ÷ 30 = 30 ÷ 30 which means C = 1 making A = 8×C = 8 × 1 = 8 B = 7×C = 7 × 1 = 7 D = 5×C = 5 × 1 = 5 E = 6×C = 6 × 1 = 6 F = 3×C = 3 × 1 = 3 and ABCDEF = 871563