Puzzle for January 7, 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
Add C and D to both sides of eq.3: D - C + C + D = E - D + C + D which becomes eq.3a) 2×D = E + C Add F and C to both sides of eq.4: E - F + F + C = D + F - C + F + C which becomes eq.4a) E + C = D + 2×F
Hint #2
In eq.4a, replace E + C with 2×D (from eq.3a): 2×D = D + 2×F Subtract D from each side of the equation above: 2×D - D = D + 2×F - D which makes D = 2×F
Hint #3
In eq.3a, replace D with (2×F): 2×(2×F) = E + C which becomes eq.3b) 4×F = E + C
Hint #4
In eq.5, replace B + C with A + E (from eq.2): E + F = A + E - E which becomes eq.5a) E + F = A
Hint #5
Add A and C to both sides of eq.6: B + E - A + A + C = A - C + D + F + A + C which becomes eq.6a) B + E + C = 2×A + D + F
Hint #6
In eq.6a, substitute 4×F for E + C (from eq.3b), (E + F) for A (from eq.5a), and 2×F for D: B + 4×F = 2×(E + F) + 2×F + F which is equivalent to B + 4×F = 2×E + 2×F + 2×F + F which becomes B + 4×F = 2×E + 5×F Subtract 4×F from each side of the above equation: B + 4×F - 4×F = 2×E + 5×F - 4×F which becomes eq.6b) B = 2×E + F
Hint #7
Substitute 2×E + F for B (from eq.6b) in eq.5: E + F = 2×E + F + C - E which becomes E + F = E + F + C Subtract E and F from both sides of the above equation: E + F - E - F = E + F + C - E - F which makes 0 = C
Hint #8
Substitute 0 for C in eq.3b: 4×F = E + 0 which makes 4×F = E
Hint #9
Substitute 4×F for E in eq.5a: 4×F + F = A which makes 5×F = A
Hint #10
Substitute (4×F) for E in eq.6b: B = 2×(4×F) + F which becomes B = 8×F + F which makes B = 9×F
Solution
Substitute 5×F for A, 9×F for B, 0 for C, 2×F for D, and 4×F for E in eq.1: 5×F + 9×F + 0 + 2×F + 4×F + F = 21 which simplifies to 21×F = 21 Divide both sides of the above equation by 21: 21×F ÷ 21 = 21 ÷ 21 which means F = 1 making A = 5×F = 5 × 1 = 5 B = 9×F = 9 × 1 = 9 D = 2×F = 2 × 1 = 2 E = 4×F = 4 × 1 = 4 and ABCDEF = 590241