Puzzle for December 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
In eq.4, replace B with C + F (from eq.3): C + D = C + F + F which becomes C + D = C + 2×F Subtract C from each side of the equation above: C + D - C = C + 2×F - C which becomes D = 2×F
Hint #2
In eq.2, replace D with 2×F: E = 2×F + F which makse E = 3×F
Hint #3
Subtract F from each side of eq.3: B - F = C + F - F which becomes eq.3a) B - F = C
Hint #4
In eq.5, substitute 3×F for E: 3×F + F = A + B which becomes 4×F = A + B Subtract B from each side of the above equation: 4×F - B = A + B - B which becomes eq.5a) 4×F - B = A
Hint #5
Substitute 4×F - B for A (from eq.5a), 3×F for E, and B - F for C (from eq.3a) in eq.6: 4×F - B + 3×F = B + B - F + F which becomes 7×F - B = 2×B Add B to both sides of the above equation: 7×F - B + B = 2×B + B which makes 7×F = 3×B Divide both sides by 3: 7×F ÷ 3 = 3×B ÷ 3 which makes 2⅓×F = B
Hint #6
Substitute 2⅓×F for B in eq.5a: 4×F - 2⅓×F = A which makes 1⅔×F = A
Hint #7
Substitute 2⅓×F for B in eq.3a: 2⅓×F - F = C which makes 1⅓×F = C
Solution
Substitute 1⅔×F for A, 2⅓×F for B, 1⅓×F for C, 2×F for D, and 3×F for E in eq.1: 1⅔×F + 2⅓×F + 1⅓×F + 2×F + 3×F + F = 34 which simplifies to 11⅓×F = 34 Divide both sides of the above equation by 11⅓: 11⅓×F ÷ 11⅓ = 34 ÷ 11⅓ which means F = 3 making A = 1⅔×F = 1⅔ × 3 = 5 B = 2⅓×F = 2⅓ × 3 = 7 C = 1⅓×F = 1⅓ × 3 = 4 D = 2×F = 2 × 3 = 6 E = 3×F = 3 × 3 = 9 and ABCDEF = 574693