Puzzle for March 20, 2021 ( )
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
eq.3 may be written as: C + F = A + B + E + D In the above equation, replace B + E with F (from eq.2): C + F = A + F + D Subtract F from both sides: C + F – F = A + F + D – F which becomes eq.3a) C = A + D
Hint #2
In eq.1, replace A + D with C (from eq.3a): C + E = B + C Subtract C from both sides of the above equation: C + E – C = B + C – C which makes E = B
Hint #3
In eq.2, substitute B for E: B + B = F which makes 2×B = F
Hint #4
Substitute B for E in eq.5: B = (A + C) ÷ B Multiply both sides of the above equation by B: B × B = (A + C) ÷ B × B which makes eq.5a) B² = A + C
Hint #5
Substitute B for E, and 2×B for F in eq.4: B × 2×B = B + C + D which becomes eq.4a) 2×B² = B + C + D
Hint #6
eq.6 may be written as: F = (B + C + D) ÷ 3 Multiply both sides of the equation above by 3: 3 × F = 3 × (B + C + D) ÷ 3 which becomes eq.6a) 3×F = B + C + D
Hint #7
Substitute (2×B) for F, and 2×B² for B + C + D (from eq.4a) in eq.6a: 3×(2×B) = 2×B² which becomes 6×B = 2×B² Since B ≠ 0 (from eq.5), divide both sides of the above equation by 2×B: 6×B ÷ 2×B = 2×B² ÷ 2×B which makes 3 = B and makes E = B = 3 and also makes F = 2×B = 2×3 = 6
Hint #8
Substitute 6 for F, 3 for B, and A + D for C (from eq.3a) into eq.6a: 3×6 = 3 + A + D + D which becomes 18 = 3 + A + 2×D Subtract 3 and 2×D from both sides of the equation above: 18 – 3 – 2×D = 3 + A + 2×D – 3 – 2×D which becomes eq.6b) 15 – 2×D = A
Hint #9
Substitute 3 for B, and A + D for C (from eq.3a) in eq.5a: 3² = A + A + D which makes 9 = 2×A + D Subtract 2×A from each side of the equation above: 9 – 2×A = 2×A + D – 2×A which makes eq.5b) 9 – 2×A = D
Hint #10
Substitute (15 – 2×D) for A (from eq.6b) into eq.5b: 9 – 2×(15 – 2×D) = D which becomes 9 – 30 + 4×D = D which becomes –21 + 4×D = D Subtract 4×D from both sides of the above equation: –21 + 4×D – 4×D = D – 4×D which makes –21 = –3×D Divide both sides by (–3): –21 ÷ (–3) = –3×D ÷ (–3) which makes 7 = D
Hint #11
Substitute 7 for D in eq.6b: 15 – 2×7 = A which becomes 15 – 14 = A which makes 1 = A
Solution
Substitute 1 for A, and 7 for D in eq.3a: C = 1 + 7 which makes C = 8 making ABCDEF = 138736