Puzzle for June 10, 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
In eq.5, replace A with B + E (from eq.2): B + C + E = B + E + D Subtract B and E from each side of the above equation: B + C + E – B – E = B + E + D – B – E which simplifies to C = D
Hint #2
In eq.4, replace D with C: C + F = A + C Subtract C from each side of the above equation: C + F – C = A + C – C which makes F = A
Hint #3
Add B to both sides of eq.3: E – B + B = B + D + B which becomes E = 2×B + D In the above equation, substitute C for D: eq.3a) E = 2×B + C
Hint #4
Substitute 2×B + C for E (from eq.3a) into eq.2: B + 2×B + C = A which becomes eq.2a) 3×B + C = A
Hint #5
Substitute C for D, and A for F in eq.6: C + C = B – C + E + A which becomes 2×C = B – C + E + A Add C to both sides of the above equation: 2×C + C = B – C + E + A + C which makes eq.6a) 3×C = B + E + A
Hint #6
In eq.6a, substitute 2×B + C for E (from eq.3a), and 3×B + C for A (from eq.2a): 3×C = B + 2×B + C + 3×B + C which becomes 3×C = 6×B + 2×C Subtract 2×C from each side of the equation above: 3×C – 2×C = 6×B + 2×C – 2×C which makes C = 6×B and also makes D = C = 6×B
Hint #7
Substitute 6×B for C in eq.2a: 3×B + 6×B = A which makes 9×B = A and also makes F = A = 9×B
Hint #8
Substitute 6×B for C in eq.3a: E = 2×B + 6×B which makes E = 8×B
Solution
Substitute 9×B for A and F, 6×B for C and D, and 8×B for E in eq.1: 9×B + B + 6×B + 6×B + 8×B + 9×B = 39 which simplifies to 39×B = 39 Divide both sides of the above equation by 39: 39×B ÷ 39 = 39 ÷ 39 which means B = 1 making A = F = 9×B = 9 × 1 = 9 C = D = 6×B = 6 × 1 = 6 E = 8×B = 8 × 1 = 8 and ABCDEF = 916689