Puzzle for May 20, 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
Add C and F to both sides of eq.3: E – C + C + F = B – F + C + F which becomes E + F = B + C In eq.6, replace E + F with B + C: B + C + D = A + B + C Subtract B and C from each side of the above equation: B + C + D – B – C = A + B + C – B – C which makes D = A
Hint #2
eq.4 may be written as: A + D + B = C + E In the equation above, replace A + D with B + F (from eq.2): B + F + B = C + E which may be written as eq.4a) C + E = 2×B + F
Hint #3
Add the left and right sides of eq.3 to the left and right sides of eq.4a, respectively: C + E + E – C = 2×B + F + B – F which simplifies to 2×E = 3×B Divide both sides of the above equation by 2: 2×E ÷ 2 = 3×B ÷ 2 which makes E = 1½×B
Hint #4
In eq.4, substitute D for A, and 1½×B for E: D + B + D = C + 1½×B which becomes 2×D + B = C + 1½×B Subtract B from both sides of the above equation: 2×D + B – B = C + 1½×B – B which becomes eq.4b) 2×D = C + ½×B
Hint #5
Add D to both sides of eq.5: C – D + F + D = B + D + D which becomes C + F = B + 2×D In the above equation, substitute C + ½×B for 2×D (from eq.4b): C + F = B + C + ½×B which becomes C + F = 1½×B + C Subtract C from each side of the equation above: C + F – C = 1½×B + C – C which makes F = 1½×B
Hint #6
Substitute 1½×B for E and F in eq.3: 1½×B – C = B – 1½×B which becomes 1½×B – C = –½×B Add C and ½×B to both sides of the above equation: 1½×B – C + C + ½×B = –½×B + C + ½×B which makes 2×B = C
Hint #7
Substitute 2×B for C in eq.4b: 2×D = 2×B + ½×B which becomes 2×D = 2½×B Divide both sides of the above equation by 2: 2×D ÷ 2 = 2½×B ÷ 2 which makes D = 1¼×B and also makes A = D = 1¼×B
Solution
Substitute 1¼×B for A and D, 2×B for C, and 1½×B for E and F in eq.1: 1¼×B + B + 2×B + 1¼×B + 1½×B + 1½×B = 34 which simplifies to 8½×B = 34 Divide both sides of the equation above by 8½: 8½×B ÷ 8½ = 34 ÷ 8½ which means B = 4 making A = D = 1¼×B = 1¼ × 4 = 5 C = 2×B = 2 × 4 = 8 E = F = 1½×B = 1½ × 4 = 6 and ABCDEF = 548566