Puzzle for September 17, 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.
* BC is a 2-digit number (not B×C).
Scratchpad
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Hint #1
Add D to both sides of eq.4: F – D + D = B + D + D which becomes eq.4a) F = B + 2×D Subtract A, D, and F from both sides of eq.2: A + C – A – D – F = D + E + F – A – D – F which becomes eq.2a) C – D – F = E – A Subtract A and C from both sides of eq.3: C + E – A – C = A + B + D – A – C which becomes eq.3a) E – A = B + D – C
Hint #2
In eq.3a, replace E – A with C – D – F (from eq.2a): C – D – F = B + D – C Add C, D, and F to both sides of the above equation: C – D – F + C + D + F = B + D – C + C + D + F which becomes eq.3b) 2×C = B + 2×D + F
Hint #3
In eq.3b, replace B + 2×D with F (from eq.4a): 2×C = F + F which makes 2×C = 2×F Divide both sides of the equation above by 2: 2×C ÷ 2 = 2×F ÷ 2 which makes C = F
Hint #4
In eq.2, substitute A + E for D (from eq.5), and C for F: A + C = A + E + E + C which becomes A + C = A + 2×E + C Subtract A and C from each side of the equation above: A + C – A – C = A + 2×E + C – A – C which makes 0 = 2×E which means 0 = E
Hint #5
Substitute 0 for E in eq.5: D = A + 0 which makes D = A
Hint #6
eq.6 may be written as: 10×B + C = B + C + F Substitute C for F in the above equation: 10×B + C = B + C + C which becomes 10×B + C = B + 2×C Subtract C and B from both sides of the above equation: 10×B + C – C – B = B + 2×C – C – B which makes 9×B = C and also makes F = C = 9×B
Hint #7
Substitute 9×B for F in eq.4a: 9×B = B + 2×D Subtract B from each side of the above equation: 9×B – B = B + 2×D – B which makes 8×B = 2×D Divide both sides by 2: 8×B ÷ 2 = 2×D ÷ 2 which makes 4×B = D and also makes A = D = 4×B
Solution
Substitute 4×B for A and D, 9×B for C and F, and 0 for E in eq.1: 4×B + B + 9×B + 4×B + 0 + 9×B = 27 which simplifies to 27×B = 27 Divide both sides of the above equation by 27: 27×B ÷ 27 = 27 ÷ 27 which means B = 1 making A = D = 4×B = 4 × 1 = 4 C = F = 9×B = 9 × 1 = 9 and ABCDEF = 419409