Puzzle for July 6, 2021 ( )
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.6, replace B + E with A + D (from eq.2): D + F = A + A + D which becomes D + F = 2×A + D Subtract D from each side of the above equation: D + F – D = 2×A + D – D which makes F = 2×A
Hint #2
Subtract D and F from both sides of eq.3: D + E – D – F = B + F – D – F which becomes E – F = B – D Add the left and right sides of the above equation to the left and right sides of eq.5, respectively: E + F + E – F = B + D + B – D which becomes 2×E = 2×B Divide both sides of the above equation by 2: 2×E ÷ 2 = 2×B ÷ 2 which makes E = B
Hint #3
In eq.4, replace E with B: C + B = A + B Subtract B from both sides of the equation above: C + B – B = A + B – B which makes C = A
Hint #4
In eq.3, substitute B for E: D + B = B + F Subtract B from each side of the above equation: D + B – B = B + F – B which makes D = F and makes D = F = 2×A
Hint #5
Substitute 2×A for D and F, and B for E in eq.6: 2×A + 2×A = A + B + B which becomes 4×A = A + 2×B Subtract A from both sides of the above equation: 4×A – A = A + 2×B – A which becomes 3×A = 2×B Divide both sides by 2: 3×A ÷ 2 = 2×B ÷ 2 which makes 1½×A = B and also makes E = B = 1½×A
Solution
Substitute 1½×A for B and E, A for C, and 2×A for D and F in eq.1: A + 1½×A + A + 2×A + 1½×A + 2×A = 36 which simplifies to 9×A = 36 Divide both sides of the above equation by 9: 9×A ÷ 9 = 36 ÷ 9 which means A = 4 making B = E = 1½×A = 1½ × 4 = 6 C = A = 4 D = F = 2×A = 2 × 4 = 8 and ABCDEF = 464868