Puzzle for February 9, 2020  ( )

Scratchpad

Find the 6-digit number ABCDEF by solving the following equations:

eq.1) A + B + C + D + E + F = 31 eq.2) E + F = A + B + C eq.3) D = E – B eq.4) A – F = E – D eq.5) B + C = A – B eq.6) D×F – B×D = A + C

A, B, C, D, E, and F each represent a one-digit positive integer.

Scratchpad

 

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Hint #1


Add D and F to both sides of eq.4: A – F + D + F = E – D + D + F which becomes eq.4a) A + D = E + F   In eq.2, replace E + F with A + D (from eq.4a): A + D = A + B + C Subtract A from both sides of the above equation: A + D – A = A + B + C – A which becomes eq.2a) D = B + C


  

Hint #2


In eq.3, substitute B + C for D (from eq.2a): B + C = E – B Add B to each side of the above equation: B + C + B = E – B + B which becomes eq.3a) 2×B + C = E


  

Hint #3


Add B to both sides of eq.5: B + C + B = A – B + B which becomes eq.5a) 2×B + C = A   In eq.5a, substitute E for 2×B + C (from eq.3a): E = A


  

Hint #4


Substitute A for E in eq.4a: A + D = A + F Subtract A from both sides of the equation above: A + D – A = A + F – A which makes D = F


  

Hint #5


Subtract B from both sides of eq.2: E + F – B = A + B + C – B which becomes E + F – B = A + C which is the same as F + E – B = A + C Substitute D for F, and D for E – B (from eq.3) in the above equation: D + D = A + C which becomes 2×D = A + C Subtract C from both sides: 2×D – C = A + C – C which makes 2×D – C = A and also makes eq.2b) E = A = 2×D – C


  

Hint #6


Subtract B from both sides of eq.2a: D – B = B + C – B which becomes eq.2c) D – B = C   Substitute D for F, and 2×D – C for A (from eq.2b) in eq.6: D×D – B×D = 2×D – C + C which becomes D × (D – B) = 2×D In the equation above, replace (D – B) with C (from eq.2c): D × C = 2×D Divide both sides by D: D × C ÷ D = 2×D ÷ D which makes C = 2


  

Hint #7


Substitute 2 for C in eq.2a: D = B + 2 Subtract 2 from each side of the above equation: D – 2 = B + 2 – 2 which makes eq.2d) D – 2 = B


  

Hint #8


Substitute 2 for C in eq.2b: eq.2e) E = A = 2×D – 2


  

Solution

Substitute 2×D – 2 for A and E (from eq.2e), D – 2 for B (from eq.2d), 2 for C, and D for F in eq.1: 2×D – 2 + D – 2 + 2 + D + 2×D – 2 + D = 31 which simplifies to 7×D – 4 = 31 Add 4 to both sides of the equation above: 7×D – 4 + 4 = 31 + 4 which becomes 7×D = 35 Divide both sides by 7: 7×D ÷ 7 = 35 ÷ 7 which means D = 5 making A = E = 2×D – 2 = 2×5 – 2 = 10 – 2 = 8 (from eq.2e) B = D – 2 = 5 – 2 = 3 (from eq.2d) F = D = 5 and ABCDEF = 832585