Puzzle for August 5, 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.
Scratchpad
Help Area
Hint #1
In eq.6, replace F with C + E (from eq.2): E + C + E = C + D – E which becomes 2×E + C = C + D – E In the above equation, subtract C from both sides, and add E to both sides: 2×E + C – C + E = C + D – E – C + E which becomes 3×E = D
Hint #2
Add E to both sides of eq.6: E + F + E = C + D – E + E which becomes eq.6a) 2×E + F = C + D In eq.5, replace D with 3×E: 3×E – E + F = A + B which becomes eq.5a) 2×E + F = A + B
Hint #3
In eq.5a, replace 2×E + F with C + D (from eq.6a): eq.5b) C + D = A + B
Hint #4
Subtract the left and right sides of eq.5b from the left and right sides of eq.3, respectively: A + C – (C + D) = B + D – (A + B) which becomes A + C – C – D = B + D – A – B which becomes A – D = D – A Add A and D to both sides of the above equation: A – D + A + D = D – A + A + D which makes 2×A = 2×D Divide both sides by 2: 2×A ÷ 2 = 2×D ÷ 2 which makes A = D and also makes A = D = 3×E
Hint #5
In eq.3, substitute A for D: A + C = B + A Subtract A from both sides of the above equation: A + C – A = B + A – A which makes C = B
Hint #6
Substitute C for B, 3×E for A and D, and (C + E) for F (from eq.2) in eq.4: C + C = 3×E + 3×E – (C + E) which becomes 2×C = 6×E – C – E which becomes 2×C = 5×E – C Add C to both sides of the above equation: 2×C + C = 5×E – C + C which makes 3×C = 5×E Divide both sides by 3: 3×C ÷ 3 = 5×E ÷ 3 which makes C = 1⅔×E and also makes B = C = 1⅔×E
Hint #7
Substitute 1⅔×E for C in eq.2: 1⅔×E + E = F which makes 2⅔×E = F
Solution
Substitute 3×E for A and D, 1⅔×E for B and C, and 2⅔×E for F in eq.1: 3×E + 1⅔×E + 1⅔×E + 3×E + E + 2⅔×E = 39 which simplifies to 13×E = 39 Divide both sides of the above equation by 13: 13×E ÷ 13 = 39 ÷ 13 which means E = 3 making A = D = 3×E = 3 × 3 = 9 B = C = 1⅔×E = 1⅔ × 3 = 5 F = 2⅔×E = 2⅔ × 3 = 8 and ABCDEF = 955938