Puzzle for May 22, 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
Subtract C and F from each side of eq.5: B + C – C – F = A + D + F – C – F which becomes eq.5a) B – F = A + D – C eq.4 may be re-written as: C – D = B – F + E In the equation above, replace B – F with A – C + D (from eq.5a): C – D = A – C + D + E Add C to both sides, and subtract D from both sides: C – D + C – D = A – C + D + E + C – D which becomes 2×C – 2×D = A + E which may be written as eq.4a) 2×(C – D) = A + E
Hint #2
Subtract A and D from both sides of eq.2: D + E – A – D = A + C – A – D which becomes eq.2a) E – A = C – D In eq.4a, substitute E – A for C – D (from eq.2a): 2×(E – A) = A + E which becomes 2×E – 2×A = A + E In the equation above, add 2×A to each side, and subtract E from each side: 2×E – 2×A + 2×A – E = A + E + 2×A – E which makes E = 3×A
Hint #3
Subtract B and D from both sides of eq.5: B + C – B – D = A + D + F – B – D which becomes C – D = A + F – B Substitute E – A for C – D (from eq.2a) in the above equation: E – A = A – B + F Add A and B to each side of the equation above: E – A + A + B = A – B + F + A + B which becomes eq.5b) E + B = 2×A + F
Hint #4
Substitute 3×A for E in eq.5b: 3×A + B = 2×A + F Subtract 2×A and B from both sides of the equation above: 3×A + B – 2×A – B = 2×A + F – 2×A – B which becomes eq.5c) A = F – B
Hint #5
Substitute F – B for A (from eq.5c) in eq.3: F – B – F = B which becomes –B = B Add B to both sides of the above equation: –B + B = B + B which makes 0 = 2×B which means 0 = B
Hint #6
Substitute 0 for B in eq.5c: A = F – 0 which makes A = F
Hint #7
Substitute 3×A for E, and A for F in eq.6: 3×A + A = D – 3×A which becomes 4×A = D – 3×A Add 3×A to each side of the equation above: 4×A + 3×A = D – 3×A + 3×A which makes 7×A = D
Hint #8
Substitute 7×A for D, and 3×A for E in eq.2: 7×A + 3×A = A + C which becomes 10×A = A + C Subtract A from each side of the equation above: 10×A – A = A + C – A which makes 9×A = C
Solution
Substitute 0 for B, 9×A for C, 7×A for D, 3×A for E, and A for F in eq.1: A + 0 + 9×A + 7×A + 3×A + A = 21 which simplifies to 21×A = 21 Divide both sides of the equation above by 21: 21×A ÷ 21 = 21 ÷ 21 which means A = 1 making C = 9×A = 9 × 1 = 9 D = 7×A = 7 × 1 = 7 E = 3×A = 3 × 1 = 3 F = A = 1 and ABCDEF = 109731