Puzzle for October 28, 2022 ( )
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
eq.4 may be written as: A + D = B + F + C In the above equation, replace B + F with D + E (from eq.3): A + D = D + E + C Subtract D from each side of the equation above: A + D - D = C + D + E - D which becomes eq.4a) A = C + E
Hint #2
In eq.2, replace C + E with A (from eq.4a): A = A + B Subtract A from each side of the above equation: A - A = A + B - A which makes 0 = B
Hint #3
In eq.3, substitute 0 for B: D + E = 0 + F which becomes D + E = F Subtract E from each side of the equation above: D + E - E = F - E which becomes eq.3a) D = F - E
Hint #4
Substitute 0 for B, and F - E for D (from eq.3a) in eq.6: 0 + C - E = F - E + E + F which becomes C - E = 2×F Add E to both sides of the above equation: C - E + E = 2×F + E which becomes eq.6a) C = 2×F + E
Hint #5
Substitute 2×F + E for C (from eq.6a) in eq.4a: A = 2×F + E + E which becomes eq.4b) A = 2×F + 2×E
Hint #6
Substitute 2×F + 2×E for A (from eq.4b), and 0 for B in eq.5: 2×F + 2×E - F = 0 - E + F which becomes F + 2×E = -E + F In the equation above, subtract F from both sides, and add E to both sides: F + 2×E - F + E = -E + F - F + E which makes 3×E = 0 which means E = 0
Hint #7
Substitute 0 for E in eq.4b: A = 2×F + 2×0 which becomes A = 2×F + 0 which makes A = 2×F
Hint #8
Substitute 0 for E in eq.6a: C = 2×F + 0 which makes C = 2×F
Hint #9
Substitute 0 for E in eq.3a: D = F - 0 which makes D = F
Solution
Substitute 2×F for A and C, 0 for B and E, and F for D in eq.1: 2×F + 0 + 2×F + F + 0 + F = 6 which simplifies to 6×F = 6 Divide both sides of the above equation by 6: 6×F ÷ 6 = 6 ÷ 6 which means F = 1 making A = C = 2×F = 2 × 1 = 2 D = F = 1 and ABCDEF = 202101