Puzzle for May 27, 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 the left and right sides of eq.2 from the left and right sides of eq.1, respectively: A - C = B + D + E - (B + E + F) which becomes A - C = B + D + E - B - E - F which becomes eq.1a) A - C = D - F
Hint #2
In eq.3, replace A - C with D - F (from eq.1a): E - F = D - F Add F to both sides of the above equation: E - F + F = D - F + F which makes E = D
Hint #3
In eq.4, replace E with D: A - F = F - (D - D) which becomes A - F = F - 0 which becomes A - F = F Add F to both sides of the equation above: A - F + F = F + F which makes A = 2×F
Hint #4
In eq.5, substitute 2×F for A: D = 2×F ÷ F which makes D = 2 and also makes D = E = 2
Hint #5
Substitute 2×F for A, and 2 for D and E in eq.1: 2×F = B + 2 + 2 which becomes 2×F = B + 4 Subtract 4 from each side of the equation above: 2×F - 4 = B + 4 - 4 which becomes eq.1b) 2×F - 4 = B
Hint #6
Substitute 2×F - 4 for B (from eq.1b), and 2 for E in eq.2: C = 2×F - 4 + 2 + F which becomes eq.2a) C = 3×F - 2
Hint #7
Substitute 2×F - 4 for B (from eq.1b), 2 for E and D, 2×F for A, and (3×F - 2) for C (from eq.2a) in eq.6: 2×F - 4 + 2 = (2×F × F) - ((3×F - 2) × 2) which becomes 2×F - 2 = 2×F² - (6×F - 4) which becomes 2×F - 2 = 2×F² - 6×F + 4 In the above equation, subtract 2×F from both sides, and add 2 to both sides: 2×F - 2 - 2×F + 2 = 2×F² - 6×F + 4 - 2×F + 2 which becomes 0 = 2×F² - 8×F + 6 Divide both sides by 2: 0 = (2×F² - 8×F + 6) ÷ 2 which becomes eq.6a) 0 = F² - 4×F + 3
Hint #8
eq.6a is a quadratic equation in standard form. The quadratic equation solution formula could be used to solve for F in eq.6a. However, eq.6a can be easily factored into the product of two expressions: (F - 1) × (F - 3) = 0 The above equation makes either: (F - 1) = 0 which means F = 1 or: (F - 3) = 0 which means F = 3
Hint #9
Check: F = 1 ... Substituting 1 for F in eq.1b would yield: 2×1 - 4 = B which would become 2 - 4 = B which would make -2 = B Since B is non-negative, then: -2 ≠ B which means F ≠ 1 and therefore makes F = 3
Solution
Since F = 3, then: A = 2×F = 2×3 = 6 B = 2×F - 4 = 2×3 - 4 = 6 - 4 = 2 (from eq.1b) C = 3×F - 2 = 3×3 - 2 = 9 - 2 = 7 (from eq.2a) and ABCDEF = 627223