Puzzle for May 26, 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
In eq.5, replace A + D with B + C (from eq.3): B + E + F = B + C - E Subtract B from both sides of the above equation: B + E + F - B = B + C - E - B which becomes E + F = C - E Add E to both sides, and subtract F from both sides: E + F + E - F = C - E + E - F which becomes eq.5a) 2×E = C - F
Hint #2
Add both B and F to each side of eq.4: C - B + B + F = D + E - F + B + F which becomes C + F = D + E + B which may be written as C + F = D + B + E In the above equation, replace B + E with D (from eq.2): C + F = D + D which becomes C + F = 2×D which may be written as eq.4a) 2×D = C + F
Hint #3
Add the left and right sides of eq.5a to the left and right sides of eq.4a, respectively: 2×D + 2×E = C + F + C - F which becomes 2×D + 2×E = 2×C Divide both sides of the equation above by 2: (2×D + 2×E) ÷ 2 = 2×C ÷ 2 which becomes eq.4b) D + E = C
Hint #4
In eq.4, substitute C for D + E (from eq.4a): C - B = C - F Subtract C from both sides: C - B - C = C - F - C which makes -B = -F which means B = F
Hint #5
Substitute B + E for D (from eq.2) in eq.3: A + B + E = B + C Subtract B from each side: A + B + E - B = B + C - B which becomes A + E = C Substitute D + E for C (from eq.4b) in the above equation: A + E = D + E Subtract E from each side: A + E - E = D + E - E which makes A = D
Hint #6
Substitute A for D in eq.2: A = B + E Subtract B from each side: A - B = B + E - B which becomes eq.2a) A - B = E Substitute A for D in eq.3: A + A = B + C Subtract B from each side: A + A - B = B + C - B which becomes eq.3a) 2×A - B = C
Hint #7
Substitute 2×A - B for C (from eq.3a), A for D, (A - B) for E (from eq.2a), and B for F in eq.6: 2×A - B + A + (A - B) = A + B + B which becomes 4×A - 2×B = A + 2×B In the above equation, add 2×B to each side, and subtract A from each side: 4×A - 2×B + 2×B - A = A + 2×B + 2×B - A which simplifies to 3×A = 4×B Divide both sides by 4: 3×A ÷ 4 = 4×B ÷ 4 which makes ¾×A = B
Solution
Substitute (2×A - B) for C (from eq.3a), A for D, (A - B) for E (from eq.2a), and B for F in eq.1: A + B + (2×A - B) + A + (A - B) + B = 20 which simplifies to 5×A = 20 Divide both sides by 5: 5×A ÷ 5 = 20 ÷ 5 which means A = 4 making F = B = ¾×A = ¾ × 4 = 3 C = 2×A - B = 2×4 - 3 = 8 - 3 = 5 (from eq.3a) D = A = 4 E = A - B = 4 - 3 = 1 (from eq.2a) and ABCDEF = 435413