Puzzle for February 21, 2020 ( )
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.4, respectively: D + E – (D + F) = A + B – (B + C) which is equivalent to D + E – D – F = A + B – B – C which becomes E – F = A – C In the equation above, add C to both sides, and subtract E from each side: E – F + C – E = A – C + C – E which becomes –F + C = A – E which may be written as eq.4a) C – F = A – E
Hint #2
In eq.3, replace A – E with C – F (from eq.4a): C + F = C – F In the above equation, add F to both sides, and subtract C from each side: C + F + F – C = C – F + F – C which makes 2×F = 0 which means F = 0
Hint #3
In eq.5, replace F with 0: E + 0 = D – E Add E to both sides of the equation above: E + 0 + E = D – E + E which makes 2×E = D
Hint #4
In eq.3, substitute 0 for F, and add E to both sides: C + 0 + E = A – E + E which becomes eq.3a) C + E = A
Hint #5
Substitute C + E for A (from eq.3a) in eq.6: B – C + E = C + E + C – E which becomes B – C + E = 2×C Add C to both sides of the equation above: B – C + E + C = 2×C + C which becomes eq.6a) B + E = 3×C
Hint #6
Substitute 2×E for D, and 0 for F in eq.2: 2×E + 0 = B + C which becomes 2×E = B + C Subtract C from each side of the above equation: 2×E – C = B + C – C which becomes eq.2a) 2×E – C = B
Hint #7
Substitute 2×E – C for B (from eq.2a) in eq.6a: 2×E – C + E = 3×C which becomes 3×E – C = 3×C Add C to both sides of the equation above: 3×E – C + C = 3×C + C which makes 3×E = 4×C Divide both sides by 4: 3×E ÷ 4 = 4×C ÷ 4 which makes ¾×E = C
Hint #8
Substitute ¾×E for C in eq.3a: ¾×E + E = A which makes 1¾×E = A
Hint #9
Substitute ¾×E for C in eq.2a: 2×E – ¾×E = B which makes 1¼×E = B
Solution
Substitute 1¾×E for A, 1¼×E for B, ¾×E for C, 2×E for D, and 0 for F in eq.1: 1¾×E + 1¼×E + ¾×E + 2×E + E + 0 = 27 which simplifies to 6¾×E = 27 Divide both sides of the equation above by 6¾: 6¾×E ÷ 6¾ = 27 ÷ 6¾ which means E = 4 making A = 1¾×E = 1¾ × 4 = 7 B = 1¼×E = 1¼ × 4 = 5 C = ¾×E = ¾ × 4 = 3 D = 2×E = 2 × 4 = 8 and ABCDE4 = 753840