Puzzle for April 3, 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.
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Hint #1
In eq.6, substitute (B + E) for D (from eq.2): E - B = B - (B + E) which becomes E - B = B - B - E which becomes E - B = -E Add B and E to both sides of the above equation: E - B + B + E = -E + B + E which makes 2×E = B
Hint #2
In eq.2, replace B with 2×E: D = 2×E + E which makes D = 3×E
Hint #3
In eq.3, replace D with 3×E: F = 3×E + E which makes F = 4×E
Hint #4
In eq.5, replace D with 3×E, and F with 4×E: A = 3×E + E + 4×E which makes A = 8×E
Hint #5
In eq.4, substitute 8×E for A: C = 8×E + E which makes C = 9×E
Solution
Substitute 8×E for A, 2×E for B, 9×E for C, 3×E for D, and 4×E for F in eq.1: 8×E + 2×E + 9×E + 3×E + E + 4×E = 27 which simplifies to 27×E = 27 Divide both sides of the above equation by 27: 27×E ÷ 27 = 27 ÷ 27 which means E = 1 making A = 8×E = 8 × 1 = 8 B = 2×E = 2 × 1 = 2 C = 9×E = 9 × 1 = 9 D = 3×E = 3 × 1 = 3 F = 4×E = 4 × 1 = 4 and ABCDEF = 829314