Puzzle for November 10, 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 positive integer.
Scratchpad
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Hint #1
eq.5 may be written as: A + C + E + D = B – D + F In the above equation, replace A + C + E with F (from eq.2): F + D = B – D + F Subtract F from both sides, and add D to both sides: F + D – F + D = B – D + F – F + D which simplifies to 2×D = B
Hint #2
In eq.3, replace B with 2×D: C + D = A + 2×D – D which becomes C + D = A + D Subtract D from each side of the equation above: C + D – D = A + D – D which makes C = A
Hint #3
In eq.4, substitute C for A, and C + F for B + D (from eq.1): E + F = C + C + F – E In the above equation, subtract F from both sides, and add E to both sides: E + F – F + E = C + C + F – E – F + E which simplifies to 2×E = 2×C Divide both sides by 2: 2×E ÷ 2 = 2×C ÷ 2 which makes E = C
Hint #4
In eq.2, substitute C for A and E: F = C + C + C which makes eq.2a) F = 3×C
Hint #5
Substitute 2×D for B, and 3×C for F in eq.1: 2×D + D = C + 3×C which makes 3×D = 4×C Divide both sides of the above equation by 4: 3×D ÷ 4 = 4×C ÷ 4 which makes ¾×D = C and also makes C = A = E = ¾×D
Hint #6
Substitute (¾×D) for C in eq.2a: F = 3×(¾×D) which makes F = 2¼×D
Solution
Substitute ¾×D for C and E, and 2¼×D for F in eq.6: ¾×D × D = ¾×D + 2¼×D which makes ¾×D² = 3×D Divide both sides of the above equation by ¾×D: ¾×D² ÷ ¾×D = 3×D ÷ ¾×D which means D = 4 making A = C = E = ¾×D = ¾ × 4 = 3 B = 2×D = 2 × 4 = 8 F = 2¼×D = 2¼ × 4 = 9 and ABCDEF = 383439