Puzzle for October 11, 2019 ( )
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 A to both sides of eq.4: F + A = E – A + A which becomes F + A = E In eq.2, replace E with F + A: B + D + F = F + A Subtract F from both sides of the above equation: B + D + F – F = F + A – F which becomes eq.2a) B + D = A
Hint #2
In eq.5, replace A with B + D (from eq.2a): D = B + D + B which becomes D = 2×B + D Subtract D from both sides of the equation above: D – D = 2×B + D – D which makes 0 = 2×B which means 0 = B
Hint #3
In eq.2a, replace B with 0: 0 + D = A which means D = A
Hint #4
In eq.2, substitute 0 for B: 0 + D + F = E which becomes eq.2b) D + F = E
Hint #5
Substitute D for A, and 0 for B in eq.6: C – D + F = D + 0 + D which becomes C – D + F = 2×D In the above equation, add D to both sides, and subtract F from each side: C – D + F + D – F = 2×D + D – F which becomes eq.6a) C = 3×D – F
Hint #6
Add C to both sides of eq.3: E – C + C = C – A + C which becomes E = 2×C – A Substitute D + F for E (from eq.2b), (3×D – F) for C (from eq.6a), and D for A in the equation above: D + F = 2×(3×D – F) – D which is equivalent to D + F = 6×D – 2×F – D which becomes D + F = 5×D – 2×F Add 2×F to both sides, and subtract D from both sides: D + F + 2×F – D = 5×D – 2×F + 2×F – D which makes 3×F = 4×D Divide both sides by 4: 3×F ÷ 4 = 4×D ÷ 4 which makes ¾×F = D and also makes A = D = ¾×F
Hint #7
Substitute ¾×F for D in eq.2b: ¾×F + F = E which makes 1¾×F = E
Hint #8
Substitute ¾×F for D in eq.6a: C = 3×(¾×F) – F which becomes C = 2¼×F – F which makes C = 1¼×F
Solution
Substitute ¾×F for A and D, 0 for B, 1¼×F for C, and 1¾×F for E in eq.1: ¾×F + 0 + 1¼×F + ¾×F + 1¾×F + F = 22 which simplifies to 5½×F = 22 Divide both sides of the equation above by 5½: 5½×F ÷ 5½ = 22 ÷ 5½ which means F = 4 making A = D = ¾×F = ¾ × 4 = 3 C = 1¼×F = 1¼ × 4 = 5 E = 1¾×F = 1¾ × 4 = 7 and ABCDEF = 305374