Puzzle for January 9, 2023 ( )
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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.
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Hint #1
In eq.2, substitute B - E for A (from eq.3), and (B + E) for F (from eq.4): E = B - E - (B + E) which becomes E = B - E - B - E which makes E = -2×E Add 2×E to both sides of the above equation: E + 2×E = -2×E + 2×E which makes 3×E = 0 which means E = 0
Hint #2
In eq.3, replace E with 0: A = B - 0 which makes A = B
Hint #3
In eq.4, replace E with 0: F = B + 0 which makes F = B
Hint #4
In eq.5, substitute B for F: B = C - B Add B to both sides of the equation above: B + B = C - B + B which makes 2×B = C
Hint #5
Substitute B for F, and 2×B for C in eq.6: D - B = 2×B + B which becomes D - B = 3×B Add B to each side of the above equation: D - B + B = 3×B + B which makes D = 4×B
Solution
Substitute B for A and F, 2×B for C, 4×B for D, and 0 for E in eq.1: B + B + 2×B + 4×B + 0 + B = 18 which simplifies to 9×B = 18 Divide both sides of the above equation by 9: 9×B ÷ 9 = 18 ÷ 9 which means B = 2 making A = F = B = 2 C = 2×B = 2 × 2 = 4 D = 4×B = 4 × 2 = 8 and ABCDEF = 224802