Puzzle for December 5, 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
In eq.1, replace E + F with C + D (from eq.4): D + C + D = B + C – F which becomes C + 2×D = B + C – F Subtract C from both sides of the above equation: C + 2×D – C = B + C – F – C which becomes eq.1a) 2×D = B – F
Hint #2
Add the left and right sides of eq.3 to the left and right sides of eq.2, respectively: E + A = B + D + D + E which becomes E + A = B + 2×D + E Subtract E from each side of the equation above: E + A – E = B + 2×D + E – E which becomes eq.2a) A = B + 2×D
Hint #3
In eq.2a, replace 2×D with B – F (from eq.1a): A = B + B – F which becomes A = 2×B – F Add F to both sides of the equation above: A + F = 2×B – F + F which becomes eq.2b) A + F = 2×B
Hint #4
In eq.6, substitute 2×B for A + F (from eq.2b): B × D = 2×B Divide both sides of the equation above by B: B × D ÷ B = 2×B ÷ B which makes D = 2
Hint #5
Substitute 2 for D in eq.3: A = 2 + E Subtract 2 from each side of the above equation: A – 2 = 2 + E – 2 which becomes eq.3a) A – 2 = E
Hint #6
Substitute 2 for D in eq.5: A × 2 = B + C + E which may be written as eq.5a) 2×A = B + C + E Add E to both sides of eq.1: D + E + F + E = B + C – F + E which becomes eq.1b) D + 2×E + F = B + C + E – F
Hint #7
Substitute 2 for D, (A – 2) for E (from eq.3a), and 2×A for B + C + E (from eq.5a) in eq.1b: 2 + 2×(A – 2) + F = 2×A – F which becomes 2 + 2×A – 4 + F = 2×A – F which becomes 2×A – 2 + F = 2×A – F Subtract 2×A from both sides of the equation above: 2×A – 2 + F – 2×A = 2×A – F – 2×A which becomes –2 + F = –F Add 2 and F to each side: –2 + F + 2 + F = –F + 2 + F which makes 2×F = 2 Divide both sides by 2: 2×F ÷ 2 = 2 ÷ 2 which makes F = 1
Hint #8
Substitute 2 for D, and 1 for F in eq.1a: 2×2 = B – 1 which becomes 4 = B – 1 Add 1 to both sides of the equation above: 4 + 1 = B – 1 + 1 which makes 5 = B
Hint #9
Substitute 5 for B, and 2 for D in eq.2a: A = 5 + 2×2 which becomes A = 5 + 4 which makes A = 9
Hint #10
Substitute 9 for A in eq.3a: 9 – 2 = E which makes 7 = E
Solution
Substitute 2 for D, 7 for E, and 1 for F in eq.4: C + 2 = 7 + 1 Subtract 2 from each side of the above equation: C + 2 – 2 = 7 + 1 – 2 which makes C = 6 and ABCDEF = 956271