Puzzle for August 6, 2021 ( )
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.
* AB, DE, and EF are 2-digit numbers (not A×B, D×E, or E×F).
Scratchpad
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Hint #1
Add (C – B) to both sides of eq.4: F + (C – B) = (E – A) – (C – B) + (C – B) which becomes eq.4a) F + C – B = E – A Subtract A and C from both sides of eq.2: C + E – F – A – C = A + B – A – C which becomes E – F – A = B – C which may be written as eq.2a) E – A – F = B – C
Hint #2
In eq.2a, replace E – A with F + C – B (from eq.4a): F + C – B – F = B – C which becomes C – B = B – C Add B and C to both sides of the above equation: C – B + B + C = B – C + B + C which simplifies to 2×C = 2×B Divide both sides by 2: 2×C ÷ 2 = 2×B ÷ 2 which makes C = B
Hint #3
eq.5 may be written as: 10×A + B = C + 10×E + F Subtract B and 10×E from both sides of the above equation: 10×A + B – B – 10×E = C + 10×E + F – B – 10×E which becomes 10×A – 10×E = C + F – B which is the same as eq.5a) 10×A – 10×E = F + C – B
Hint #4
In eq.5a, change F + C – B to E – A (from eq.4a): 10×A – 10×E = E – A Add 10×E and A to both sides of the equation above: 10×A – 10×E + 10×E + A = E – A + 10×E + A which becomes 11×A = 11×E Divide both sides by 11: 11×A ÷ 11 = 11×E ÷ 11 which means A = E
Hint #5
In eq.4a, substitute B for C, and A for E: F + B – B = A – A which means F = 0
Hint #6
Substitute B for C, and 0 for F in eq.3: A + D = B + 0 – A Add A to both sides of the equation above: A + D + A = B + 0 – A + A which becomes eq.3a) 2×A + D = B
Hint #7
eq.6 may be written as: 10×D + E + F = 10×A + B + D + E Subtract D and E from each side of the equation above: 10×D + E + F – D – E = 10×A + B + D + E – D – E which becomes eq.6a) 9×D + F = 10×A + B
Hint #8
Substitute 0 for F, and 2×A + D for B (from eq.3a) in eq.6a: 9×D + 0 = 10×A + 2×A + D which becomes 9×D = 12×A + D Subtract D from each side of the above equation: 9×D – D = 12×A + D – D which becomes 8×D = 12×A Divide both sides by 8: 8×D ÷ 8 = 12×A ÷ 8 which makes D = 1½×A
Hint #9
Substitute 1½×A for D in eq.3a: 2×A + 1½×A = B which makes 3½×A = B and also makes C = B = 3½×A
Solution
Substitute 3½×A for B and C, 1½×A for D, A for E, and 0 for F in eq.1: A + 3½×A + 3½×A + 1½×A + A + 0 = 21 which simplifies to 10½×A = 21 Divide both sides of the above equation by 10½: 10½×A ÷ 10½ = 21 ÷ 10½ which means A = 2 making B = C = 3½×A = 3½ × 2 = 7 D = 1½×A = 1½ × 2 = 3 E = A = 2 and ABCDEF = 277320