Puzzle for December 23, 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
Add F to both sides of eq.5: D + F + F = A + B + C - F + F which becomes eq.5a) D + 2×F = A + B + C Add E and F to both sides of eq.2: F - E + E + F = D - F + E + F which becomes eq.2a) 2×F = D + E
Hint #2
In eq.1, replace A + B + C with D + 2×F (from eq.5a), D + E with 2×F (from eq.2a): D + 2×F + 2×F + F = 26 which becomes D + 5×F = 26 Subtract 5×F from both sides of the equation above: D + 5×F - 5×F = 26 - 5×F which becomes eq.1a) D = 26 - 5×F
Hint #3
To make eq.1a true, check several possible values for F and D: If F = 4, then D = 26 - 5×4 = 26 - 20 = 6 If F = 5, then D = 26 - 5×5 = 26 - 25 = 1 If F = 6, then D = 26 - 5×6 = 26 - 30 = -4 If F > 6, then D < -4 If F = 3, then D = 26 - 5×3 = 26 - 15 = 11 If F < 3, then D > 11 Since D and F must be one-digit non-negative integers, the above equations make either: F = 4 and D = 6 or F = 5 and D = 1
Hint #4
Check: F = 5, and D = 1 ... Substituting 1 for D, and 5 for F in eq.3 would yield: B + E = 1 - 5 which would make B + E = -4 Since B and E must be non-negative, then: B + E ≥ 0 which means B + E ≠ -4 Therefore: F ≠ 5 and D ≠ 1 which means: F = 4 and D = 6
Hint #5
In eq.2, substitute 4 for F, and 6 for D: 4 - E = 6 - 4 which becomes 4 - E = 2 In the above equation, add E to both sides, and subtract 2 from both sides: 4 - E + E - 2 = 2 + E - 2 which makes 2 = E
Hint #6
Substitute 2 for E, 6 for D, and 4 for F in eq.3: B + 2 = 6 - 4 which becomes B + 2 = 2 Subtract 2 from both sides of the above equation: B + 2 - 2 = 2 - 2 which makes B = 0
Hint #7
Substitute 6 for D, 4 for F, and 0 for B in eq.5: 6 + 4 = A + 0 + C - 4 which becomes 10 = A + C - 4 Add 4 to both sides of the equation above: 10 + 4 = A + C - 4 + 4 which becomes eq.5b) 14 = A + C
Hint #8
Substitute 6 for D, 0 for B, and 2 for E in eq.4: C + 6 = A - 0 + 2 which becomes C + 6 = A + 2 Subtract 2 from each side of the equation above: C + 6 - 2 = A + 2 - 2 which becomes eq.4a) C + 4 = A
Hint #9
Substitute C + 4 for A (from eq.4a) into eq.5b: 14 = C + 4 + C which becomes 14 = 2×C + 4 Subtract 4 from each side of the equation above: 14 - 4 = 2×C + 4 - 4 which makes 10 = 2×C Divide both sides by 2: 10 ÷ 5 = 2×C ÷ 2 which makes 5 = C
Solution
Substitute 5 for C in eq.4a: 5 + 4 = A which makes 9 = A and makes ABCDEF = 905624