Puzzle for November 18, 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 positive integer.
Scratchpad
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Hint #1
In eq.5, replace A + C with B + D (from eq.3): D + F = B + D Subtract D from each side of the equation above: D + F – D = B + D – D which makes B = F
Hint #2
Subtract the left and right sides of eq.5 from the left and right sides of eq.4, respectively: C + F – (D + F) = A + D – (A + C) which becomes C + F – D – F = A + D – A – C which becomes C – D = D – C Add D and C to both sides of the above equation: C – D + D + C = D – C + D + C which makes 2×C = 2×D Divide both sides by 2: 2×C ÷ 2 = 2×D ÷ 2 which makes C = D
Hint #3
Subtract the left and right sides of eq.4 from the left and right sides of eq.3, respectively: A + C – (C + F) = B + D – (A + D) which becomes A + C – C – F = B + D – A – D which becomes A – F = B – A Add F and A to both sides of the above equation: A – F + F + A = B – A + F + A which becomes eq.3a) 2×A = B + F
Hint #4
In eq.3a, replace F with B: 2×A = B + B which makes 2×A = 2×B Divide both sides of the above equation by 2: 2×A ÷ 2 = 2×B ÷ 2 which makes A = B
Hint #5
In eq.6, substitute B for A and F, and C for D: C × E = B + C + C + B which becomes eq.6a) C × E = 2×B + 2×C
Hint #6
Subtract C from each side of eq.2: E – C = B + C – C which becomes eq.2a) E – C = B
Hint #7
Substitute (E – C) for B (from eq.2a) into eq.6a: C × E = 2×(E – C) + 2×C which becomes C × E = 2×E – 2×C + 2×C which makes C × E = 2×E Divide both sides of the above equation by E: C × E ÷ E = 2×E ÷ E which makes C = 2 and also makes C = D = 2
Hint #8
Substitute 2 for C in eq.2: eq.2b) E = B + 2
Hint #9
Substitute B for A and F, 2 for C and D, and B + 2 for E (from eq.2b) in eq.1: B + B + 2 + 2 + B + 2 + B = 34 which simplifies to 4×B + 6 = 34 Subtract 6 from both sides of the above equation: 4×B + 6 – 6 = 34 – 6 which makes 4×B = 28 Divide both sides by 4: 4×B ÷ 4 = 28 ÷ 4 which means B = 7
Solution
Since B = 7, then: A = F = B = 7 E = B + 2 = 7 + 2 = 9 (from eq.2b) and ABCDEF = 772297