Puzzle for July 30, 2021 ( )
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
Subtract the left and right sides of eq.5 from the left and right sides of eq.4, respectively: B + E – (E – B) = A + D + F – (A + B + D) which becomes B + E – E + B = A + D + F – A – B – D which becomes 2×B = F – B Add B to both sides of the above equation: 2×B + B = F – B + B which becomes 3×B = F
Hint #2
In eq.2, replace A + C with D + F (from eq.3): E = D + F + F which becomes eq.2a) E = D + 2×F
Hint #3
In eq.2a, substitute (3×B) for F: E = D + 2×(3×B) which becomes eq.2b) E = D + 6×B
Hint #4
Substitute D + 6×B for E (from eq.2b) in eq.5: D + 6×B – B = A + B + D which becomes D + 5×B = A + B + D Subtract D and B from both sides of the above equation: D + 5×B – D – B = A + B + D – D – B which makes 4×B = A
Hint #5
Substitute D + F for A + C (from eq.3) in eq.6: B – D + E = D + F + D which becomes B – D + E = 2×D + F Add D to both sides of the above equation: B – D + E + D = 2×D + F + D which becomes eq.6a) B + E = 3×D + F
Hint #6
Substitute 3×D + F for B + E (from eq.6a), and 4×B for A in eq.4: 3×D + F = 4×B + D + F Subtract F and D from both sides of the equation above: 3×D + F – F – D = 4×B + D + F – F – D which becomes 2×D = 4×B Divide both sides by 2: 2×D ÷ 2 = 4×B ÷ 2 which makes D = 2×B
Hint #7
Substitute 2×B for D in eq.2b: E = 2×B + 6×B which makes E = 8×B
Hint #8
Substitute 2×B for D, 3×B for F, and 4×B for A in eq.3: 2×B + 3×B = 4×B + C which becomes 5×B = 4×B + C Subtract 4×B from both sides of the equation above: 5×B – 4×B = 4×B + C – 4×B which makes B = C
Solution
Substitute 4×B for A, B for C, 2×B for D, 8×B for E, and 3×B for F in eq.1: 4×B + B + B + 2×B + 8×B + 3×B = 19 which simplifies to 19×B = 19 Divide both sides of the above equation by 19: 19×B ÷ 19 = 19 ÷ 19 which means B = 1 making A = 4×B = 4 × B = 4 C = B = 1 D = 2×B = 2 × 1 = 2 E = 8×B = 8 × 1 = 8 F = 3×B = 3 × 1 = 3 and ABCDEF = 411283