Puzzle for January 21, 2022 ( )
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.6, subtract D from both sides, and add C to both sides: B + D – E + F – D + C = A – C – D + E – D + C which becomes B – E + F + C = A – 2×D + E which may be written as eq.6a) B + C – E + F = A – 2×D + E
Hint #2
In eq.6a, replace B + C – E + F with A – B (from eq.4): A – B = A – 2×D + E In the above equation, subtract A from both sides, and add B and 2×D to both sides: A – B – A + B + 2×D = A – 2×D + E – A + B + 2×D which becomes 2×D = E + B which may be written as eq.6b) 2×D = B + E
Hint #3
In eq.2, replace B + E with 2×D (from eq.6b): D + F = 2×D Subtract D from each side of the above equation: D + F – D = 2×D – D which makes F = D
Hint #4
Add C to both sides of eq.6: B + D – E + F + C = A – C – D + E + C which becomes B + C + D – E + F = A – D + E In the above equation, substitute A – C + E for B + C + D (from eq.3): A – C + E – E + F = A – D + E which becomes eq.6c) A – C + F = A – D + E
Hint #5
Substitute D for F in eq.6c: A – C + D = A – D + E In the equation above, subtract A from both sides, and add C and D to both sides: A – C + D – A + C + D = A – D + E – A + C + D which simplifies to eq.6d) 2×D = E + C
Hint #6
Substitute B + E for 2×D (from eq.6b) into eq.6d: B + E = E + C Subtract E from each side of the equation above: B + E – E = E + C – E which makes B = C
Hint #7
Substitute B for C, and D for F in eq.5: B + D + E = B – E + D Subtract B and D from both sides of the above equation: B + D + E – B – D = B – E + D – B – D which simplifies to E = –E Add E to both sides: E + E = –E + E which makes 2×E = 0 which means E = 0
Hint #8
Substitute 0 for E in eq.6b: 2×D = B + 0 which makes 2×D = B and also makes 2×D = B = C
Hint #9
Substitute 2×D for B and C, and 0 for E in eq.3: 2×D + 2×D + D = A – 2×D + 0 which becomes 5×D = A – 2×D Add 2×D to both sides of the above equation: 5×D + 2×D = A – 2×D + 2×D which makes 7×D = A
Solution
Substitute 7×D for A, 2×D for B and C, 0 for E, and D for F in eq.1: 7×D + 2×D + 2×D + D + 0 + D = 13 which simplifies to 13×D = 13 Divide both sides of the above equation by 13: 13×D ÷ 13 = 13 ÷ 13 which means D = 1 making A = 7×D = 7×1 = 7 B = C = 2×D = 2×1 = 2 F = D = 1 and ABCDEF = 722101