Puzzle for October 29, 2022 ( )
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.4 may be written as: E + F = A + C + B In the equation above, replace A + C with E (from eq.2): E + F = E + B Subtract E from each side: E + F - E = E + B - E which makes F = B
Hint #2
In eq.3, replace F with B: A + B = B + D Subtract B from each side of the above equation: A + B - B = B + D - B which makes A = D
Hint #3
eq.5 may be written as: A + D = (B + C + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × (A + D) = 3 × (B + C + F) ÷ 3 which becomes eq.5a) 3×A + 3×D = B + C + F
Hint #4
In eq.5a, substitute D for A, and B for F: 3×D + 3×D = B + C + B which becomes 6×D = 2×B + C Subtract 2×B from both sides of the equation above: 6×D - 2×B = 2×B + C - 2×B which becomes eq.5b) 6×D - 2×B = C
Hint #5
eq.6 may be written as: B - D = (C + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × (B - D) = 3 × (C + E + F) ÷ 3 which becomes eq.6a) 3×B - 3×D = C + E + F
Hint #6
Substitute D for A in eq.2: eq.2a) E = D + C
Hint #7
Substitute D + C for E (from eq.2a), and B for F in eq.6a: 3×B - 3×D = C + D + C + B which becomes 3×B - 3×D = 2×C + D + B Subtract D and B from both sides of the above equation: 3×B - 3×D - D - B = 2×C + D + B - D - B which becomes eq.6b) 2×B - 4×D = 2×C
Hint #8
Substitute (6×D - 2×B) for C (from eq.5b) in eq.6b: 2×B - 4×D = 2×(6×D - 2×B) which becomes 2×B - 4×D = 12×D - 4×B Add 4×D and 4×B to both sides of the equation above: 2×B - 4×D + 4×D + 4×B = 12×D - 4×B + 4×D + 4×B which makes 6×B = 16×D Divide both sides by 16: 6×B ÷ 16 = 16×D ÷ 16 which makes ⅜×B = D and also makes ⅜×B = D = A
Hint #9
Substitute (⅜×B) for D in eq.5b: 6×(⅜×B) - 2×B = C which becomes 2¼×B - 2×B = C which makes ¼×B = C
Hint #10
Substitute ⅜×B for D, and ¼×B for C in eq.2a: E = ⅜×B + ¼×B which makes E = ⅝×B
Solution
Substitute ⅜×B for A and D, ¼×B for C, ⅝×B for E, and B for F in eq.1: ⅜×B + B + ¼×B + ⅜×B + ⅝×B + B = 29 which simplifies to 3⅝×B = 29 Divide both sides of the above equation by 3⅝: 3⅝×B ÷ 3⅝ = 29 ÷ 3⅝ which means B = 8 making A = D = ⅜×B = ⅜ × 8 = 3 C = ¼×B = ¼ × 8 = 2 E = ⅝×B = ⅝ × 8 = 5 F = B = 8 and ABCDEF = 382358