Puzzle for July 18, 2020 ( )
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
Add E and C to both sides of eq.3: F – E + E + C = B – C + E + C which becomes F + C = B + E which is the same as eq.3a) C + F = B + E Add D to both sides of eq.5: C – D + F + D = A + D + E + D which becomes eq.5a) C + F = A + 2×D + E
Hint #2
In eq.5a, replace C + F with B + E (from eq.3a): B + E = A + 2×D + E Subtract E from each side of the above equation: B + E – E = A + 2×D + E – E which becomes eq.5b) B = A + 2×D
Hint #3
In eq.5b, substitute C + D for B (from eq.4): C + D = A + 2×D Subtract D from both sides of the above equation: C + D – D = A + 2×D – D which becomes eq.5c) C = A + D
Hint #4
Subtract the left and right sides of eq.5 from the left and right sides of eq.2, respectively: A + C – (C – D + F) = E + F – (A + D + E) which is equivalent to A + C – C + D – F = E + F – A – D – E which becomes A + D – F = F – A – D Add A, D, and F to both sides of the equation above: A + D – F + A + D + F = F – A – D + A + D + F which simplifies to 2×A + 2×D = 2×F Divide both sides by 2: (2×A + 2×D) ÷ 2 = 2×F ÷ 2 which becomes eq.2a) A + D = F
Hint #5
Substitute C for A + D (from eq.5c) in eq.2a: C = F
Hint #6
Substitute C for F in eq.2: A + C = E + C Subtract C from each side of the equation above: A + C – C = E + C – C which becomes A = E
Hint #7
Substitute F for C in eq.3a: F + F = B + E which becomes eq.3b) 2×F = B + E
Hint #8
eq.1 may be written as: A + D + B + E + C + F = 35 Substitute F for A + D (from eq.2a), 2×F for B + E (from eq.3b), and F for C in the equation above: F + 2×F + F + F = 35 which makes 5×F = 35 Divide both sides by 5: 5×F ÷ 5 = 35 ÷ 5 which makes F = 7 and also makes C = F = 7
Hint #9
Substitute A for E, and 7 for C in eq.6: A×B – A = 7×A + A which becomes A×B – A = 8×A Add A to both sides of the above equation: A×B – A + A = 8×A + A which becomes A×B = 9×A Divide both sides by A: A×B ÷ A = 9×A ÷ A which makes B = 9
Hint #10
Substitute 7 for F, and 9 for B in eq.3b: 2×7 = 9 + E which becomes 14 = 9 + E Subtract 9 from each side of the above equation: 14 – 9 = 9 + E – 9 which makes 5 = E and also makes A = E = 5
Solution
Substitute 7 for C, and 5 for A in eq.5c: 7 = 5 + D Subtract 5 from both sides of the equation above: 7 – 5 = 5 + D – 5 which makes 2 = D making ABCDEF = 597257