Puzzle for August 14, 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.
* DE is a 2-digit number (not D×E).
Scratchpad
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Hint #1
Subtract C from both sides of eq.3: B + C + D – C = A + E + F – C which becomes B + D = A + E + F – C which may be written as B + D = A – C + E + F In the above equation, replace A – C with E – F (from eq.4): B + C + D = E – F + C + E + F which becomes B + C + D = C + 2×E Subract C from each side: B + C + D – C = C + 2×E – C which becomes eq.3a) B + D = 2×E
Hint #2
eq.5 may be written as: 10×D + E = B + D + C In the above equation, replace B + D with 2×E (from eq.3a): 10×D + E = 2×E + C Subtract E from each side: 10×D + E – E = 2×E + C – E which becomes eq.5a) 10×D = E + C
Hint #3
Add F and C to each side of eq.4: E – F + F + C = A – C + F + C which makes E + C = A + F and also makes eq.4a) 10×D = E + C = A + F (from eq.5a)
Hint #4
eq.1 may be written as: A + F + B + D + E + C = 28 Substitute 10×D for both A + F and E + C (from eq.4a) in the above equation: 10×D + B + D + 10×D = 28 which becomes B + 21×D = 28 Subtract 21×D from both sides: B + 21×D – 21×D = 28 – 21×D which becomes eq.1a) B = 28 – 21×D
Hint #5
To make eq.1a true, check several possible values for D and B: If D = 0, then B = 28 – 21×0 = 28 – 0 = 28 If D = 1, then B = 28 – 21×1 = 28 – 21 = 7 If D = 2, then B = 28 – 21×2 = 28 – 42 = –14 If D > 2, then B < –14 Since B must be a one-digit non-negative integer, then B = 7 which makes D = 1
Hint #6
Substitute 7 for B, and 1 for D in eq.3a: 7 + 1 = 2×E which makes 8 = 2×E Divide both sides of the equation above by 2: 8 ÷ 2 = 2×E ÷ 2 which makes 4 = E
Hint #7
Substitute 1 for D, and 4 for E in eq.5a: 10×1 = 4 + C which becomes 10 = 4 + C Subtract 4 from each side of the above equation: 10 – 4 = 4 + C – 4 which makes 6 = C
Hint #8
Substitute 1 for D, and 4 for E in eq.2: F = 1 + 4 which makes F = 5
Solution
Substitute 4 for E, 5 for F, and 6 for C in eq.4: 4 – 5 = A – 6 which becomes –1 = A – 6 Add 6 to both sides of the above equation: –1 + 6 = A – 6 + 6 which makes 5 = A making ABCDEF = 576145