Puzzle for June 6, 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
Add E to both sides of eq.2: D - E + E = A + E + E which becomes eq.2a) D = A + 2×E Add F to both sides of eq.4: F - E + F = C - F + F which becomes eq.4a) 2×F - E = C
Hint #2
In eq.6, replace C with 2×F - E (from eq.4a), and D with A + 2×E (from eq.2a): A + F = 2×F - E + A + 2×E - F which becomes A + F = F + E + A Subtract A and F from each side of the equation above: A + F - A - F = F + E + A - A - F which simplifies to 0 = E
Hint #3
In eq.3, substitute 0 for E: B - 0 = A - B which becomes B = A - B Add B to both sides of the above equation: B + B = A - B + B which becomes 2×B = A
Hint #4
In eq.2a, substitute 2×B for A, and 0 for E: D = 2×B + 2×0 which becomes D = 2×B + 0 which makes D = 2×B
Hint #5
Substitute 0 for E, and 2×B for A in eq.5: C + 0 = 2×B + B - C which becomes C = 3×B - C Add C to both sides of the equation above: C + C = 3×B - C + C which makes 2×C = 3×B Divide both sides by 2: 2×C ÷ 2 = 3×B ÷ 2 which makes C = 1½×B
Hint #6
Substitute 0 for E, and 1½×B for C in eq.4a: 2×F - 0 = 1½×B which makes 2×F = 1½×B Divide both sides of the above equation by 1½: 2×F ÷ 1½ = 1½×B ÷ 1½ which makes F = ¾×B
Solution
Substitute 2×B for A and D, 1½×B for C, 0 for E, and ¾×B for F in eq.1: 2×B + B + 1½×B + 2×B + 0 + ¾×B = 29 which simplifies to 7¼×B = 29 Divide both sides of the above equation by 7¼: 7¼×B ÷ 7¼ = 29 ÷ 7¼ which means B = 4 making A = D = 2×B = 2 × 4 = 8 C = 1½×B = 1½ × 4 = 6 F = ¾×B = ¾ × 4 = 3 and ABCDEF = 846803