Puzzle for June 26, 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.
Scratchpad
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Hint #1
In eq.4, replace F with A + D (from eq.2), and C with A - D (from eq.3): D + A + D = A - D - D which becomes 2×D + A = A - 2×D In the above equation, subtract A from both sides, and add D to both sides: 2×D + A - A + 2×D = A - 2×D - A + 2×D which becomes 4×D = 0 which means D = 0
Hint #2
In eq.2, substitute 0 for D: F = A + 0 which makes F = A
Hint #3
In eq.3, substitute 0 for D: C = A - 0 which makes C = A and also makes C = A = F
Hint #4
In eq.5, substitute 0 for D: F - B = B - 0 which becomes F - B = B Add B to both sides of the above equation: F - B + B = B + B which makes F = 2×B and also makes C = A = F = 2×B
Hint #5
Substitute 0 for D, and 2×B for A and C and F in eq.6: 0 + E - B = 2×B + B + 2×B - 0 + 2×B which becomes E - B = 7×B Add B to both sides of the above equation: E - B + B = 7×B + B which makes E = 8×B
Solution
Substitute 2×B for A and C and F, 0 for D, and 8×B for E in eq.1: 2×B + B + 2×B + 0 + 8×B + 2×B = 15 which simplifies to 15×B = 15 Divide both sides of the above equation by 15: 15×B ÷ 15 = 15 ÷ 15 which means B = 1 making A = C = F = 2×B = 2 × 1 = 2 E = 8×B = 8 × 1 = 8 and ABCDEF = 212082