Puzzle for August 4, 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
Subtract F from each side of eq.3: B + F – F = E – F – F which becomes eq.3a) B = E – 2×F
Hint #2
In eq.4, substitute (E + F) for C (from eq.2), and (E – 2×F) for B (from eq.3a): (E + F) – (E – 2×F) + F = (E – 2×F) + E – (E + F) – F which becomes E + F – E + 2×F + F = 2×E – 2×F – E – F – F which becomes 4×F = E – 4×F Add 4×F to both sides of the above equation: 4×F + 4×F = E – 4×F + 4×F which makes 8×F = E
Hint #3
In eq.2, replace E with 8×F: C = 8×F + F which makes C = 9×F
Hint #4
In eq.3a, replace E with 8×F: B = 8×F – 2×F which makes B = 6×F
Hint #5
eq.5 may be written as: E = (A + B + C) ÷ 3 Multiply both sides of the above equation by 3: 3 × E = 3 × (A + B + C) ÷ 3 which becomes eq.5a) 3×E = A + B + C
Hint #6
Substitute (8×F) for E, 6×F for B, and 9×F for C in eq.5a: 3×(8×F) = A + 6×F + 9×F which becomes 24×F = A + 6×F + 9×F which becomes 24×F = A + 15×F Subtract 15×F from both sides of the equation above: 24×F – 15×F = A + 15×F – 15×F which makes 9×F = A
Hint #7
Substitute 9×F for A and C, 6×F for B, and 8×F for E in eq.1: 9×F + 6×F + 9×F + D + 8×F + F = 33 which becomes 33×F + D = 33 Subtract 33×F from each side of the above equation: 33×F + D – 33×F = 33 – 33×F which becomes eq.1a) D = 33 – 33×F
Hint #8
Since F and D must be one-digit non-negative integers, the above equations make: F = 1 and D = 0
Solution
Since F = 1, then: A = C = 9×F = 9 × 1 = 9 B = 6×F = 6 × 1 = 6 E = 8×F = 8 × 1 = 8 and ABCDEF = 969081