Puzzle for May 24, 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.
* DE is a 2-digit number (not D×E).
Scratchpad
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Hint #1
Add C, D, and E to both sides of eq.3: B - D - E + C + D + E = A - C + F + C + D + E which becomes eq.3a) B + C = A + D + E + F
Hint #2
eq.1 may be re-written as: B + C + A + D + E + F = 36 In the equation above, replace A + D + E + F with B + C (from eq.3a): B + C + B + C = 36 which is equivalent to 2×(B + C) = 36 Divide both sides by 2: 2×(B + C) ÷ 2 = 36 ÷ 2 which makes eq.1a) B + C = 18
Hint #3
Since B and C must both be one-digit non-negative integers, eq.1a makes: B = 9 and C = 9
Hint #4
In eq.3a, replace D + E + F with B (from eq.2): B + C = A + B Subtract B from each side: B + C - B = A + B - B which makes C = A and which means A = C = 9
Hint #5
eq.5 may be written as: 10×D + E + F = A + C which may also be written as 9×D + D + E + F = A + C Substitute B for D + E + F (from eq.2) in the above equation: 9×D + B = A + C Substitute 9 for A, B, and C: 9×D + 9 = 9 + 9 Subtract 9 from each side: 9×D + 9 - 9 = 9 + 9 - 9 which becomes 9×D = 9 Divide both sides by 9: 9×D ÷ 9 = 9 ÷ 9 which means D = 1
Hint #6
Substitute 9 for A and B in eq.4: 9 + E = 9 + F Subtract 9 from each side: 9 + E - 9 = 9 + F - 9 which makes E = F
Solution
Substitute 9 for B, 1 for D, and E for F in eq.2: 9 = 1 + E + E Subtract 1 from each side: 9 - 1 = 1 + E + E - 1 which makes 8 = 2×E Divide both sides by 2: 8 ÷ 2 = 2×E ÷ 2 which makes 4 = E and which also makes F = E = 4 and makes ABCDEF = 999144