Puzzle for May 29, 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
Add E and F to both sides of eq.3: D – E + E + F = A – F + E + F which becomes eq.3a) D + F = A + E In eq.2, replace D + F with A + E (from eq.3a): B + A + E = A + C + E Subtract A and E from each side of the above equation: B + A + E – A – E = A + C + E – A – E which simplifies to B = C
Hint #2
Add A to both sides of eq.4: E + A = B – A + A which becomes E + A = B which may be written as A + E = B In the above equation, replace A + E with D + F (from eq.3a): eq.4a) D + F = B
Hint #3
In eq.5, substitute (B – A) for E (from eq.4): C – A = A – (B – A) Add A to both sides of the equation above: C – A + A = A – (B – A) + A which becomes C = A – B + A + A which becomes eq.5a) C = 3×A – B
Hint #4
Substitute B for C in eq.5a: B = 3×A – B Add B to both sides of the above equation: B + B = 3×A – B + B which makes 2×B = 3×A Divide both sides by 2: 2×B ÷ 2 = 3×A ÷ 2 which makes B = 1½×A and also makes C = B = 1½×A
Hint #5
Substitute 1½×A for B in eq.4: E = 1½×A – A which makes E = ½×A
Hint #6
eq.1 may be re-written as: A + B + C + E + D + F = 36 Substitute B for C, and B for D + F (from eq.4a) in the above equation: A + B + B + E + B = 36 which becomes eq.1a) A + 3×B + E = 36
Hint #7
Substitute (1½×A) for B, and ½×A for E in eq.1a: A + 3×(1½×A) + ½×A = 36 which becomes A + 4½×A + ½×A = 36 which becomes 6×A = 36 Divide both sides of the equation above by 6: 6×A ÷ 6 = 36 ÷ 6 which makes A = 6 making C = B = 1½×A = 1½ × 6 = 9 E = ½×A = ½ × 6 = 3
Hint #8
Substitute B for C, and B for D + F (from eq.4a) in eq.6: B × E × F = B + B + B which becomes B × E × F = 3×B Substitute 9 for B, and 3 for E in the equation above: 9 × 3 × F = 3×9 which makes 27×F = 27 Divide both sides by 27: 27 ÷ 27 = 27×F ÷ 27 which means 1 = F
Solution
Substitute 1 for F, and 9 for B in eq.4a: D + 1 = 9 Subtract 1 from both sides of the above equation: D + 1 – 1 = 9 – 1 which makes D = 8 making ABCDEF = 699831