Puzzle for June 17, 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
Add B and F to both sides of eq.5: A – B + C + B + F = B + E – F + B + F which becomes eq.5a) A + C + F = 2×B + E eq.6 may be written as: E = (A + B + C + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × E = 4 × (A + B + C + F) ÷ 4 which becomes 4×E = A + B + C + F which may be written as eq.6a) 4×E = B + A + C + F
Hint #2
In eq.6a, replace A + C + F with 2×B + E (from eq.5a): 4×E = B + 2×B + E which becomes 4×E = 3×B + E Subtract E from each side of the above equation: 4×E – E = 3×B + E – E which makes 3×E = 3×B Divide both sides by 3: 3×E ÷ 3 = 3×B ÷ 3 which makes E = B
Hint #3
In eq.2, replace E with B: B = D + B Subtract B from both sides of the equation above: B – B = D + B – B which makes 0 = D
Hint #4
In eq.4, substitute 0 for D, and B for E: C – 0 + B = B – C + F which becomes C + B = B – C + F In the above equation, subtract B from both sides, and add C to both sides: C + B – B + C = B – C + F – B + C which makes 2×C = F
Hint #5
In eq.3, substitute 0 for D, 2×C for F, and B for E: 0 + 2×C = A + B – 2×C which becomes 2×C = A + B – 2×C Add 2×C to each side of the above equation: 2×C + 2×C = A + B – 2×C + 2×C which becomes 4×C = A + B which may be written as eq.3a) 4×C = B + A
Hint #6
Substitute 4×C for B + A (from eq.3a), and 2×C for F in eq.6a: 4×E = 4×C + C + 2×C which becomes 4×E = 7×C Divide both sides of the equation above by 4: 4×E ÷ 4 = 7×C ÷ 4 which makes E = 1¾×C and also makes B = E = 1¾×C
Hint #7
Substitute 1¾×C for B in eq.3a: 4×C = 1¾×C + A Subtract 1¾×C from each side of the equation above: 4×C – 1¾×C = 1¾×C + A – 1¾×C which makes 2¼×C = A
Solution
Substitute 2¼×C for A, 1¾×C for B and E, 0 for D, and 2×C for F in eq.1: 2¼×C + 1¾×C + C + 0 + 1¾×C + 2×C = 35 which simplifies to 8¾×C = 35 Divide both sides of the above equation by 8¾: 8¾×C ÷ 8¾ = 35 ÷ 8¾ which means C = 4 making A = 2¼×C = 2¼ × 4 = 9 B = E = 1¾×C = 1¾ × 4 = 7 F = 2×C = 2 × 4 = 8 and ABCDEF = 974078