Puzzle for December 3, 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 non-negative integer.
Scratchpad
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Hint #1
In eq.5, replace E with A + D (from eq.2): B + D = A + D – D – F which becomes B + D = A – F Add F to each side of the equation above: B + D + F + F = A – F + F which becomes eq.2a) B + D + F = A
Hint #2
In eq.4, replace A with B + D + F (from eq.2a): B + D + F = C + D + F Subtract D and F from both sides of the above equation: C + D + F – D – F = B + D + F – D – F which simplifies to C = B
Hint #3
In eq.3, substitute B for C: B + E – F = A + B In the equation above, subtract B from both sides, and add F to both sides: B + E – F – B + F = A + B – B + F which becomes eq.3a) E = A + F
Hint #4
Substitute A + F for E (from eq.3a) in eq.2: A + F = A + D Subtract A from both sides of the equation above: A + F – A = A + D – A which makes F = D
Hint #5
Substitute D for F in eq.5: B + D = E – D – D which becomes B + D = E – 2×D Add 2×D to both sides of the above equation: B + D + 2×D = E – 2×D + 2×D which becomes eq.5a) B + 3×D = E
Hint #6
Substitute B + 3×D for E (from eq.5a), D for F, and B for C in eq.6: D + B + 3×D + D = B + B – D which becomes B + 5×D = 2×B – D In the equation above, subtract B from both sides, and add D to both sides: B + 5×D – B + D = 2×B – D – B + D which makes 6×D = B and also makes C = B = 6×D
Hint #7
Substitute 6×D for B in eq.5a: 6×D + 3×D = E which makes 9×D = E
Hint #8
Substitute 6×D for B, and D for F in eq.2a: 6×D + D + D = A which makes 8×D = A
Solution
Substitute 8×D for A, 6×D for B and C, 9×D for E, and D for F in eq.1: 8×D + 6×D + 6×D + D + 9×D + D = 31 which simplifies to 31×D = 31 Divide each side of the above equation by 31: 31×D ÷ 31 = 31 ÷ 31 which makes D = 1 making A = 8×D = 8 × 1 = 8 B = C = 6×D = 6 × 1 = 6 E = 9×D = 9 × 1 = 9 F = D = 1 and ABCDEF = 866191