Puzzle for January 23, 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.
Scratchpad
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Hint #1
In eq.4, replace A with E + F (from eq.5): B + F = E + F – C In the above equation, subtract F from both sides,and add C to both sides: B + F – F + C = E + F – C – F + C which becomes eq.4a) B + C = E
Hint #2
In eq.3, replace E with B + C (from eq.4a): C + F = B + C Subtract C from each side of the equation above: C + F – C = B + C – C which makes F = B
Hint #3
In eq.2, substitute B for F: A – B = D – B Add B to each side of the above equation: A – B + B = D – B + B which makes A = D
Hint #4
In eq.6, substitute F for B: A + F – C = C + E – F Add C and F to both sides of the equation above: A + F – C + C + F = C + E – F + C + F which becomes eq.6a) A + 2×F = 2×C + E
Hint #5
Substitute E + F for A (from eq.5) in eq.6a: E + F + 2×F = 2×C + E which becomes E + 3×F = 2×C + E Subtract E from both sides of the above equation: E + 3×F – E = 2×C + E – E which makes 3×F = 2×C Divide both sides by 2: 3×F ÷ 2 = 2×C ÷ 2 which makes 1½×F = C
Hint #6
Substitute 1½×F for C in eq.3: 1½×F + F = E which means 2½×F = E
Hint #7
Substitute 2½×F for E in eq.5: 2½×F + F = A which makes 3½×F = A and also makes D = A = 3½×F
Solution
Substitute 3½×F for A and D, F for B, 1½×F for C, and 2½×F for E in eq.1: 3½×F + F + 1½×F + 3½×F + 2½×F + F = 26 which simplifies to 13×F = 26 Divide both sides of the equation above by 13: 13×F ÷ 13 = 26 ÷ 13 which means F = 2 making A = D = 3½×F = 3½ × 2 = 7 B = F = 2 C = 1½×F = 1½ × 2 = 3 E = 2½×F = 2½ × 2 = 5 and ABCDEF = 723752