Puzzle for July 23, 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
Help Area
Hint #1
In eq.6, replace A + C with B + E + F (from eq.5): E + F = B + E + F + D – E which becomes E + F = B + F + D Subtract F from each side of the equation above: E + F – F = B + F + D – F which becomes eq.6a) E = B + D
Hint #2
In eq.4, substitute (B + D) for E (from eq.6a): D – B = C – D – (B + D) which becomes D – B = C – D – B – D which becomes D – B = C – 2×D – B Add B and 2×D to both sides of the above equation: D – B + B + 2×D = C – 2×D – B + B + 2×D which simplifies to 3×D = C
Hint #3
Substitute 3×D for C in eq.2: eq.2a) 3×D + F = A + E
Hint #4
Add B, D, and E to both sides of eq.4: D – B + B + D + E = C – D – E + B + D + E which becomes 2×D + E = C + B which may be written as eq.4a) 2×D + E = B + C
Hint #5
In eq.4a, replace B + C with A – D + F (from eq.3): 2×D + E = A – D + F Add D to both sides of the equation above: 2×D + E + D = A – D + F + D which becomes eq.4b) 3×D + E = A + F
Hint #6
Subtract the left and right sides of eq.4b from the left and right sides of eq.2a, respectively: 3×D + F – (3×D + E) = A + E – (A + F) which becomes 3×D + F – 3×D – E = A + E – A – F which becomes F – E = E – F Add E and F to both sides of the above equation: F – E + E + F = E – F + E + F which makes 2×F = 2×E Divide both sides by 2: 2×F ÷ 2 = 2×E ÷ 2 which makes F = E
Hint #7
In eq.2, substitute 3×D for C, and E for F: 3×D + E = A + E Subtract E from each side of the above equation: 3×D + E – E = A + E – E which makes 3×D = A
Hint #8
Substitute E for F, and 3×D for A and C in eq.6: E + E = 3×D + 3×D + D – E which becomes 2×E = 7×D – E Add E to both sides of the equation above: 2×E + E = 7×D – E + E which becomes 3×E = 7×D Divide both sides by 3: 3×E ÷ 3 = 7×D ÷ 3 which makes E = 2⅓×D and also makes F = E = 2⅓×D
Hint #9
Substitute 2⅓×D for E in eq.6a: 2⅓×D = B + D Subtract D from both sides of the above equation: 2⅓×D – D = B + D – D which makes 1⅓×D = B
Solution
Substitute 3×D for A and C, 1⅓×D for B, and 2⅓×D for E and F in eq.1: 3×D + 1⅓×D + 3×D + D + 2⅓×D + 2⅓×D = 39 which simplifies to 13×D = 39 Divide both sides of the above equation by 13: 13×D ÷ 13 = 39 ÷ 13 which means D = 3 making A = C = 3×D = 3 × 3 = 9 B = 1⅓×D = 1⅓ × 3 = 4 E = F = 2⅓×D = 2⅓ × 3 = 7 and ABCDEF = 949377