Puzzle for March 3, 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
In eq.4, replace D with B + E (from eq.2): E + F = B + B + E which becomes E + F = 2×B + E Subtract E from each side of the equation above: E + F – E = 2×B + E – E which makes F = 2×B
Hint #2
eq.6 may be written as: B + D = A + E + C In the above equation, replace B + D with E + F (from eq.4), and replace A + E with F (from eq.3): E + F = F + C Subtract F from each side of the above equation: E + F – F = F + C – F which makes E = C
Hint #3
In eq.5, substitute B + E for D (from eq.2): C + B + E = A + B + F Subtract B from both sides of the equation above: C + B + E – B = A + B + F – B which becomes eq.5a) C + E = A + F
Hint #4
Substitute C for E, and 2×B for F in eq.5a: C + C = A + 2×B which becomes eq.5b) 2×C = A + 2×B
Hint #5
Substitute 2×B for F, and C for E in eq.3: eq.3a) 2×B = A + C
Hint #6
Substitute A + C for 2×B (from eq.3a) into eq.5b: 2×C = A + A + C which becomes 2×C = 2×A + C Subtract C from both sides of the above equation: 2×C – C = 2×A + C – C which makes C = 2×A and also makes E = C = 2×A
Hint #7
Substitute 2×A for C in eq.3a: 2×B = A + 2×A which becomes 2×B = 3×A Divide both sides of the above equation by 2: 2×B ÷ 2 = 3×A ÷ 2 which makes B = 1½×A
Hint #8
Substitute 2×A for C and E in eq.5a: 2×A + 2×A = A + F which becomes 4×A = A + F Subtract A from each side of the above equation: 4×A - A = A + F - A which makes 3×A = F
Hint #9
Substitute 1½×A for B, and 2×A for E in eq.2: D = 1½×A + 2×A which makes D = 3½×A
Solution
Substitute 1½×A for B, 2×A for C and E, 3½×A for D, and 3×A for F in eq.1: A + 1½×A + 2×A + 3½×A + 2×A + 3×A = 26 which simplifies to 13×A = 26 Divide both sides of the above equation by 13: 13×A ÷ 13 = 26 ÷ 13 which means A = 2 making B = 1½×A = 1½ × 2 = 3 C = E = 2×A = 2 × 2 = 4 D = 3½×A = 3½ × 2 = 7 F = 3×A = 3 × 2 = 6 and ABCDEF = 234746