Puzzle for July 21, 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
In eq.4, replace D + F with A + B (from eq.2): B + E = C + A + B Subtract B from each side of the equation above: B + E – B = C + A + B – B which becomes eq.4a) E = C + A
Hint #2
In eq.3, replace E with C + A (from eq.4a): F – A = C + A – C which becomes F – A = A Add A to both sides of the above equation: F – A + A = A + A which makes F = 2×A
Hint #3
In eq.5, substitute C + A for E (from eq.4a): D + C + A = B + C – D In the above equation, subtract C from both sides, and add D to both sides: D + C + A – C + D = B + C – D – C + D which becomes eq.5a) 2×D + A = B
Hint #4
Substitute 2×A for F, and 2×D + A for B (from eq.5a) in eq.2: D + 2×A = A + 2×D + A which becomes D + 2×A = 2×A + 2×D Subtract D and 2×A from both sides of the equation above: D + 2×A – D – 2×A = 2×A + 2×D – D – 2×A which makes 0 = D
Hint #5
Substitute 0 for D in eq.5a: 2×0 + A = B which becomes 0 + A = B which makes A = B
Hint #6
Substitute 2×A for F, and A for B in eq.6: 2×A – E = A ÷ A which becomes 2×A – E = 1 In the above equation, add E to both sides, and subtract 1 from both sides: 2×A – E + E – 1 = 1 + E – 1 which becomes eq.6a) 2×A – 1 = E
Hint #7
Substitute 2×A – 1 for E (from eq.6a) in eq.4a: 2×A – 1 = C + A Subtract A from each side of the equation above: 2×A – 1 – A = C + A – A which becomes eq.4b) A – 1 = C
Hint #8
Substitute A for B, A – 1 for C (from eq.4b), 0 for D, 2×A – 1 for E (from eq.6a), and 2×A for F in eq.1: A + A + A – 1 + 0 + 2×A – 1 + 2×A = 26 which simplifies to 7×A – 2 = 26 Add 2 to both sides of the above equation: 7×A – 2 + 2 = 26 + 2 which makes 7×A = 28 Divide both sides by 7: 7×A ÷ 7 = 28 ÷ 7 which means A = 4
Solution
Since A = 4, then: B = A = 4 C = A – 1 = 4 – 1 = 3 (from eq.4b) E = 2×A – 1 = 2×4 – 1 = 8 – 1 = 7 (from eq.6a) F = 2×A = 2×4 = 8 and ABCDEF = 443078