Puzzle for October 14, 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 B and D to both sides of eq.3: B – D + B + D = C – A – B + B + D which becomes eq.3a) 2×B = C – A + D Add B to both sides of eq.4: A – B + F + B = B + D + B which becomes eq.4a) A + F = 2×B + D
Hint #2
In eq.4a, replace 2×B with C – A + D (from eq.3a): A + F = C – A + D + D which becomes A + F = C – A + 2×D Add A to both sides of the above equation: A + F + A = C – A + 2×D + A which becomes eq.4b) 2×A + F = C + 2×D
Hint #3
Subtract the left and right sides of eq.2 from the left and right sides of eq.4b, respectively: 2×A + F – (D + F) = C + 2×D – (A + C) which becomes 2×A + F – D – F = C + 2×D – A – C which becomes 2×A – D = 2×D – A Add D and A to both sides of the above equation: 2×A – D + D + A = 2×D – A + D + A which makes 3×A = 3×D Divide both sides by 3: 3×A ÷ 3 = 3×D ÷ 3 which makes A = D
Hint #4
In eq.2, substitute A for D: A + F = A + C Subtract A from both sides of the equation above: A + F – A = A + C – A which makes F = C
Hint #5
Substitute F for C, and A for D in eq.5: F + A + E = A – E + F Subtract A and F from each side of the above equation: F + A + E – A – F = A – E + F – A – F which makes E = –E Add E to both sides: E + E = –E + E which makes 2×E = 0 which means E = 0
Hint #6
Substitute A for D in eq.3: B – A = C – A – B Add A and B to both sides of the equation above: B – A + A + B = C – A – B + A + B which makes 2×B = C and also makes F = C = 2×B
Hint #7
Add A, D, and B to both sides of eq.6: E + F – A – D + A + D + B = A – B + D + A + D + B which becomes E + F + B = 2×A + 2×D Substitute 0 for E, 2×B for F, and A for D in the above equation: 0 + 2×B + B = 2×A + 2×A which becomes 3×B = 4×A Divide both sides by 4: 3×B ÷ 4 = 4×A ÷ 4 which makes ¾×B = A and also makes D = A = ¾×B
Solution
Substitute ¾×B for A and D, 2×B for C and F, and 0 for E in eq.1: ¾×B + B + 2×B + ¾×B + 0 + 2×B = 26 which simplifies to 6½×B = 26 Divide both sides of the above equation by 6½: 6½×B ÷ 6½ = 26 ÷ 6½ which means B = 4 making A = D = ¾×B = ¾ × 4 = 3 C = F = 2×B = 2 × 4 = 8 and ABCDEF = 348308