Puzzle for August 24, 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.
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Hint #1
Add E to both sides of eq.3: A + F + E = D – E + E which becomes A + F + E = D which may be written as A + E + F = D In the above equation, replace A + E with C (from eq.1): eq.3a) C + F = D
Hint #2
In eq.2, replace D with C + F (from eq.3a): C + F = B + F Subtract F from each side of the equation above: C + F – F = B + F – F which makes C = B
Hint #3
In eq.6, substitute B for C: E = B ÷ B which makes E = 1
Hint #4
Substitute 1 for E in eq.1: C = A + 1 which also makes eq.1a) B = C = A + 1
Hint #5
Substitute (A + 1) for B and C (from eq.1a), and 1 for E in eq.4: (A + 1) + 1 = A – (A + 1) + D which becomes A + 2 = A – A – 1 + D which becomes A + 2 = –1 + D Add 1 to both sides of the above equation: A + 2 + 1 = –1 + D + 1 which becomes eq.4a) A + 3 = D
Hint #6
Substitute A + 1 for C (from eq.1a), and A + 3 for D (from eq.4a) into eq.3a: A + 1 + F = A + 3 Subtract A and 1 from each side of the above equation: A + 1 + F – A – 1 = A + 3 – A – 1 which makes F = 2
Hint #7
eq.5 may be written as: A = (B + D + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × A = 3 × (B + D + F) ÷ 3 which becomes eq.5a) 3×A = B + D + F
Hint #8
In eq.5a, substitute A + 1 for B (from eq.1a), A + 3 for D (from eq.4a), and 2 for F: 3×A = A + 1 + A + 3 + 2 which becomes 3×A = 2×A + 6 Subtract 2×A from both sides of the equation above: 3×A – 2×A = 2×A + 6 – 2×A which makes A = 6
Solution
Since A = 6, then: B = C = A + 1 = 6 + 1 = 7 (from eq.1a) D = A + 3 = 6 + 3 = 9 (from eq.4a) and ABCDEF = 677912