Puzzle for February 16, 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
Subtract C and F from both sides of eq.2: C + E - C - F = A + F - C - F which becomes eq.2a) E - F = A - C Subtract A and F from both sides of eq.3: A + E - A - F = B + F - A - F which becomes eq.3a) E - F = B - A
Hint #2
In eq.3a, replace E - F with A - C (from eq.2a): A - C = B - A Add C and A to both sides of the above equation: A - C + C + A = B - A + C + A which becomes eq.3b) 2×A = B + C
Hint #3
Subtract B and F from both sides of eq.4: C + F - B - F = B + E - B - F which becomes eq.4a) C - B = E - F
Hint #4
In eq.4a, replace E - F with A - C (from eq.2a): C - B = A - C Add C to both sides of the equation above: A - C + C = C - B + C which becomes eq.4b) A = 2×C - B
Hint #5
In eq.3b, substitute (2×C - B) for A (from eq.4b): 2×(2×C - B) = B + C which becomes 4×C - 2×B = B + C In the above equation, add 2×B to both sides, and subtract C from both sides: 4×C - 2×B + 2×B - C = B + C + 2×B - C which becomes 3×C = 3×B Divide both sides by 3: 3×C ÷ 3 = 3×B ÷ 3 which makes C = B
Hint #6
Substitute C for B in eq.4b: A = 2×C - C which makes A = C
Hint #7
Substitute A for C in eq.2: A + E = A + F Subtract A from each side of the equation above: A + E - A = A + F - A which makes E = F
Hint #8
Substitute C for B and A, and E for F in eq.5: C + C - E = C + C + E which becomes 2×C - E = 2×C + E In the above equation, subtract 2×C from both sides, and add E to both sides: 2×C - E - 2×C + E = 2×C + E - 2×C + E which makes 0 = 2×E which means 0 = E and also means 0 = E = F
Hint #9
In eq.6, substitute C for A and B, and 0 for E and F: C + C = (C × D) + 0 + 0 which becomes 2×C = C × D Divide both sides by C (assumes C ≠ 0): 2×C ÷ C = C × D ÷ C which makes 2 = D
Hint #10
Confirm: C ≠ 0 ... If C = 0, then: A = C = 0 and B = C = 0 Substituting 0 for A, B, C, E, and F in eq.1 would yield: 0 + 0 + 0 + D + 0 + 0 = 23 which would make D = 23 Since D must be a one-digit integer, then: D ≠ 23 which means C ≠ 0
Solution
Substitute C for A and B, 2 for D, and 0 for E and F in eq.1: C + C + C + 2 + 0 + 0 = 23 which simplifies to 3×C + 2 = 23 Subtract 2 from both sides of the above equation: 3×C + 2 - 2 = 23 - 2 which makes 3×C = 21 Divide both sides by 3: 3×C ÷ 3 = 21 ÷ 3 which means C = 7 making A = B = C = 7 and ABCDEF = 777200