Puzzle for August 13, 2023 ( )
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.6 may be written as: E = (B + C + D) ÷ 3 Multiply both sides of the above equation by 3: 3 × E = 3 × (B + C + D) ÷ 3 which becomes eq.6a) 3×E = B + C + D
Hint #2
Add D and E to both sides of eq.4: F - D + D + E = D - E + D + E which becomes eq.4a) F + E = 2×D Add F and 2×E to both sides of eq.3: E - F + F + 2×E = C - E + F + 2×E which becomes eq.3a) 3×E = C + F + E
Hint #3
In eq.6a, substitute C + 2×D for 3×E (from eq.3a): C + 2×D = B + C + D Subtract C and D from each side of the equation above: C + 2×D - C - D = B + C + D - C - D which simplifies to D = B
Hint #4
Substitute D for B in eq.2: D + D = A + E which becomes eq.2a) 2×D = A + E
Hint #5
Substitute A + E for 2×D (from eq.2a) into eq.4a: F + E = A + E Subtract E from each side of the above equation: F + E - E = A + E - E which makes F = A
Hint #6
eq.6a may be written as: 1½×(2×E) = B + C + D Substitute (C + F) for 2×E (from eq.3a), and B for D into the above equation: 1½×(C + F) = B + C + B which becomes 1½×C + 1½×F = 2×B + C Subtract C from both sides: 1½×C + 1½×F - C = 2×B + C - C which becomes eq.6b) ½×C + 1½×F = 2×B
Hint #7
Multiply both sides of eq.6b by 2: 2 × (½×C + 1½×F) = 2 × (2×B) which becomes C + 3×F = 4×B Subtract 3×F from each side of the equation above: C + 3×F - 3×F = 4×B - 3×F which becomes eq.6c) C = 4×B - 3×F
Hint #8
Substitute 4×B - 3×F for C (from eq.6c), and F for A in eq.5: 4×B - 3×F + E = F + B - E In the above equation, add 3×F and E to both sides, and subtract B from both sides: 4×B - 3×F + E + 3×F + E - B = F + B - E + 3×F + E - B which becomes eq.5a) 3×B + 2×E = 4×F
Hint #9
Substitute B for D in eq.4a: F + E = 2×B Subtract F from both sides of the equation above: F + E - F = 2×B - F which becomes eq.4b) E = 2×B - F
Hint #10
Substitute (2×B - F) for E (from eq.4b) in eq.5a: 3×B + 2×(2×B - F) = 4×F which becomes 3×B + 4×B - 2×F = 4×F which becomes 7×B - 2×F = 4×F Add 2×F to both sides of the above equation: 7×B - 2×F + 2×F = 4×F + 2×F which makes 7×B = 6×F Divide both sides by 6: 7×B ÷ 6 = 6×F ÷ 6 which makes 1⅙×B = F
Hint #11
Substitute 1⅙×B for F in eq.4b: E = 2×B - 1⅙×B which makes E = ⅚×B
Hint #12
Substitute (1⅙×B) for F in eq.6c: C = 4×B - 3×(1⅙×B) which becomes C = 4×B - 3½×B which makes C = ½×B
Hint #13
Substitute B for D, and ⅚×B for E in eq.2a: 2×B = A + ⅚×B Subtract ⅚×B from each side of the above equation: 2×B - ⅚×B = A + ⅚×B - ⅚×B which makes 1⅙×B = A
Solution
Substitute 1⅙×B for A and F, ½×B for C, B for D, and ⅚×B for E in eq.1: 1⅙×B + B + ½×B + B + ⅚×B + 1⅙×B = 34 which simplifies to 5⅔×B = 34 Divide both sides of the above equation by 5⅔: 5⅔×B ÷ 5⅔ = 34 ÷ 5⅔ which means B = 6 making A = F = 1⅙×B = 1⅙ × 6 = 7 C = ½×B = ½ × 6 = 3 D = B = 6 E = ⅚×B = ⅚ × 6 = 5 and ABCDEF = 763657