Puzzle for December 11, 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.6, replace A with B + D (from eq.3): D + F = B + D + B which becomes D + F = 2×B + D Subtract D from each side of the above equation: D + F - D = 2×B + D - D which makes F = 2×B
Hint #2
In eq.4, replace F with 2×B: B + 2×B = C - B which becomes 3×B = C - B Add B to both sides of the equation above: 3×B + B = C - B + B which makes 4×B = C
Hint #3
In eq.5, substitute 4×B for C, and 2×B for F: 4×B + 2×B = D - 2×B which becomes 6×B = D - 2×B Add 2×B to both sides of the above equation: 6×B + 2×B = D - 2×B + 2×B which makes 8×B = D
Hint #4
Substitute 4×B for C in eq.2: E = B + 4×B which makes E = 5×B
Hint #5
Substitute 8×B for D in eq.3: A = B + 8×B which makes A = 9×B
Solution
Substitute 9×B for A, 4×B for C, 8×B for D, 5×B for E, and 2×B for F in eq.1: 9×B + B + 4×B + 8×B + 5×B + 2×B = 29 which simplifies to 29×B = 29 Divide both sides of the above equation by 29: 29×B ÷ 29 = 29 ÷ 29 which means B = 1 making A = 9×B = 9 × 1 = 9 C = 4×B = 4 × 1 = 4 D = 8×B = 8 × 1 = 8 E = 5×B = 5 × 1 = 5 F = 2×B = 2 × 1 = 2 and ABCDEF = 914852