Puzzle for December 13, 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.6: A + E + + E = D - E + F + E which becomes eq.6a) A + 2×E = D + F Add F and E to both sides of eq.4: E - F + F + E = C - E + F + E which becomes eq.4a) 2×E = C + F
Hint #2
In eq.6a, replace 2×E with C + F (from eq.4a): A + C + F = D + F Subtract F from each side of the equation above: A + C + F - F = D + F - F which becomes eq.6b) A + C = D
Hint #3
In eq.3, replace D with A + C (from eq.6b): A = C + A + C which becomes A = 2×C + A Subtract A from both sides of the equation above: A - A = 2×C + A - A which makes 0 = 2×C which means 0 = C
Hint #4
In eq.4a, substitute 0 for C: 2×E = 0 + F which makes 2×E = F
Hint #5
Substitute 2×E for F in eq.2: A = E + 2×E which makes A = 3×E
Hint #6
Substitute 3×E for A, and 0 for C in eq.3: 3×E = 0 + D which makes 3×E = D
Hint #7
Substitute 2×E for F in eq.5: B - E = 2×E - B Add E and B to both sides of the above equation: B - E + E + B = 2×E - B + E + B which makes 2×B = 3×E Divide both sides by 2: 2×B ÷ 2 = 3×E ÷ 2 which makes B = 1½×E
Solution
Substitute 3×E for A and D, 1½×E for B, 0 for C, and 2×E for F in eq.1: 3×E + 1½×E + 0 + 3×E + E + 2×E = 21 which simplifies to 10½×E = 21 Divide both sides of the above equation by 10½: 10½×E ÷ 10½ = 21 ÷ 10½ which means E = 2 making A = D = 3×E = 3 × 2 = 6 B = 1½×E = 1½ × 2 = 3 F = 2×E = 2 × 2 = 4 and ABCDEF = 630624