Puzzle for March 25, 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
Add B to both sides of eq.4: E – B + F + B = A – F + B which becomes eq.4a) E + F = A + B – F eq.5 may be written as: eq.5a) A + B – F + C – D – E = E + F
Hint #2
In eq.5a, replace A + B – F with E + F (from eq.4a): E + F + C – D – E = E + F which becomes F + C – D = E + F Subtract F from each side of the equation above: F + C – D – F = E + F – F which becomes eq.5b) C – D = E
Hint #3
In eq.3, replace E with C – D (from eq.5b): C + D = C – D In the above equation, subtract C from both sides, and add D to both sides: C + D – C + D = C – D – C + D which makes 2×D = 0 which means D = 0
Hint #4
In eq.3, substitute 0 for D: C + 0 = E which makes C = E Substitute C for E in eq.4: eq.4b) C – B + F = A – F
Hint #5
Substitute A + F for C (from eq.2) in eq.4b: A + F – B + F = A – F which becomes A – B + 2×F = A – F In the equation above, add B and F to each side, and subtract A from both sides: A – B + 2×F + B + F – A = A – F + B + F – A which simplifies to 3×F = B
Hint #6
Add B and F to both sides of eq.6: B – D + F + B + F = A – B – F + B + F which becomes 2×B – D + 2×F = A Substitute (3×F) for B, and 0 for D in the above equation: 2×(3×F) – 0 + 2×F = A which becomes 6×F + 2×F = A which makes 8×F = A
Hint #7
Substitute 3×F for B, and 8×F for A in eq.4: E – 3×F + F = 8×F – F which becomes E – 2×F = 7×F Add 2×F to each side of the equation above: E – 2×F + 2×F = 7×F + 2×F which makes E = 9×F and also makes C = E = 9×F
Solution
Substitute 8×F for A, 3×F for B, 9×F for C and E, and 0 for D in eq.1: 8×F + 3×F + 9×F + 0 + 9×F + F = 30 which simplifies to 30×F = 30 Divide both sides of the above equation by 30: 30×F ÷ 30 = 30 ÷ 30 which means F = 1 making A = 8×F = 8 × 1 = 8 B = 3×F = 3 × 1 = 3 C = E = 9×F = 9 × 1 = 9 and ABCDEF = 839091