Puzzle for October 24, 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.5, replace C with A + B (from eq.2): B + A + B = A + D which becomes 2×B + A = A + D Subtract A from each side of the equation above: 2×B + A - A = A + D - A which makes 2×B = D
Hint #2
In eq.3, replace D with 2×B: F = B + 2×B which makes F = 3×B
Hint #3
In eq.4, replace D with 2×B: eq.4a) C = 2×B + E
Hint #4
In eq.6, substitute 2×B + E for C (from eq.4a), and 3×B for F: 2×B + E + 3×B = A + E - 3×B which becomes 5×B + E = A + E - 3×B In the equation above, subtract E from both sides, and add 3×B to both sides: 5×B + E - E + 3×B = A + E - 3×B - E + 3×B which becomes 8×B = A
Hint #5
Substitute 8×B for A in eq.2: C = 8×B + B which makes C = 9×B
Hint #6
Substitute 9×B for C in eq.4a: 9×B = 2×B + E Subtract 2×B from each side of the above equation: 9×B - 2×B = 2×B + E - 2×B which makes 7×B = E
Solution
Substitute 8×B for A, 9×B for C, 2×B for D, 7×B for E, and 3×B for F in eq.1: 8×B + B + 9×B + 2×B + 7×B + 3×B = 30 which simplifies to 30×B = 30 Divide both sides of the above equation by 30: 30×B ÷ 30 = 30 ÷ 30 which means B = 1 making A = 8×B = 8 × 1 = 8 C = 9×B = 9 × 1 = 9 D = 2×B = 2 × 1 = 2 E = 7×B = 7 × 1 = 7 F = 3×B = 3 × 1 = 3 and ABCDEF = 819273