Puzzle for July 15, 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.2, replace D with E – F (from eq.3): B + E = A + C + E – F Subtract E from both sides of the above equation: B + E – E = A + C + E – F – E which becomes eq.2a) B + F = A + C
Hint #2
In eq.5, replace B + F with A + C (from eq.2a): A + D = A + C Subtract A from both sides of the above equation: A + D – A = A + C – A which makes D = C and also makes eq.5a) C = D = E – F (from eq.3)
Hint #3
Add F to both sides of eq.4: E + F + F = B – F + F which becomes eq.4a) E + 2×F = B In eq.6, substitute E + 2×F for B (from eq.4a), and (E – F) for both C and D (from eq.5a): E + 2×F – (E – F) – (E – F) = (E – F) – E which is equivalent to E + 2×F – E + F – E + F = –F which becomes –E + 4×F = –F Add E and F to each side of the above equation: –E + 4×F + E + F = –F + E + F which makes 5×F = E
Hint #4
Substitute 5×F for E in eq.4a: 5×F + 2×F = B which makes 7×F = B
Hint #5
Substitute 5×F for E in eq.5a: C = D = 5×F – F which makes C = D = 4×F
Hint #6
Substitute 7×F for B, 5×F for E, and 4×F for C and D in eq.2: 7×F + 5×F = A + 4×F + 4×F which becomes 12×F = A + 8×F Subtract 8×F from each side of the above equation: 12×F – 8×F = A + 8×F – 8×F which makes 4×F = A
Solution
Substitute 4×F for A and C and D, 7×F for B, and 5×F for E in eq.1: 4×F + 7×F + 4×F + 4×F + 5×F + F = 25 which becomes 25×F = 25 Divide both sides of the above equation by 25: 25×F ÷ 25 = 25 ÷ 25 which means F = 1 making A = C = D = 4×F = 4 × 1 = 4 B = 7×F = 7 × 1 = 7 E = 5×F = 5 × 1 = 5 and ABCDEF = 474451