Puzzle for February 19, 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
In eq.2, add A to each side, and subtract C from each side: C – D + A – C = E – A + A – C which becomes –D + A = E – C which may be written as A – D = E – C In eq.3, replace A – D with E – C: C = E – C Add C to both sides of the above equation: C + C = E – C + C which makes 2×C = E Divide both sides by 2: 2×C ÷ 2 = E ÷ 2 which makes C = ½×E
Hint #2
In eq.3, add D to both sides, and subtract C from both sides: C + D – C = A – D + D – C which becomes eq.3a) D = A – C eq.5 may be written as: D + F = A – C + B In the equation above, replace A – C with D (from eq.3a): D + F = D + B Subtract D from each side: D + F – D = D + B – D which makes F = B
Hint #3
Add B to both sides of eq.4: E + B = A – B + B which becomes eq.4a) E + B = A In eq.6, substitute B for F, and E + B for A (from eq.4a): B – E + B = E + B + D + E – B which becomes 2×B – E = D + 2×E Subtract 2×E from both sides of the equation above: 2×B – E – 2×E = D + 2×E – 2×E which becomes eq.6a) 2×B – 3×E = D
Hint #4
Substitute ½×E for C, E + B for A (from eq.4a), and (2×B – 3×E) for D (from eq.6a) in eq.3: ½×E = E + B – (2×B – 3×E) which may be written as ½×E = E + B – 2×B + 3×E which becomes ½×E = –B + 4×E In the equation above, add B to each side, and subtract ½×E from both sides: ½×E + B – ½×E = –B + 4×E + B – ½×E which makes B = 3½×E and also makes F = B = 3½×E
Hint #5
Substitute (3½×E) for B in eq.6a: 2×(3½×E) – 3×E = D which becomes 7×E – 3×E = D which makes 4×E = D
Hint #6
Substitute 3½×E for B in eq.4a: E + 3½×E = A which makes 4½×E = A
Solution
Substitute 4½×E for A, 3½×E for B and F, ½×E for C, and 4×E for D in eq.1: 4½×E + 3½×E + ½×E + 4×E + E + 3½×E = 34 which simplifies to 17×E = 34 Divide both sides of the equation above by 17: 17×E ÷ 17 = 34 ÷ 17 which makes E = 2 making A = 4½×E = 4½ × 2 = 9 B = F = 3½×E = 3½ × 2 = 7 C = ½×E = ½ × 2 = 1 D = 4×E = 4 × 2 = 8 and ABCDEF = 971827