Puzzle for March 25, 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
eq.6 may be written as: D = A + B + F + E In the above equation, replace A + B + F with C (from eq.2): eq.6a) D = C + E
Hint #2
In eq.5, replace D with C + E (from eq.6a): A + C = C + E + E – C which becomes eq.5a) A + C = 2×E
Hint #3
Add A to both sides of eq.2: C + A = A + B + F + A which is equivalent to A + C = 2×A + B + F In the above equation, substitute 2×E for A + C (from eq.5a), and A + E for B + F (from eq.3): 2×E = 2×A + A + E which becomes 2×E = 3×A + E Subtract E from each side: 2×E – E = 3×A + E – E which makes E = 3×A
Hint #4
Substitute (3×A) for E in eq.5a: A + C = 2×(3×A) which becomes A + C = 6×A Subtract A from each side of the equation above: A + C – A = 6×A – A which makes C = 5×A
Hint #5
Substitute 5×A for C, and 3×A for E in eq.5: A + 5×A = D + 3×A – 5×A which becomes 6×A = D – 2×A Add 2×A to both sides of the equation above: 6×A + 2×A = D – 2×A + 2×A which makes 8×A = D
Hint #6
Substitute 3×A for E, and 5×A for C in eq.4: B + 3×A = 5×A + F Subtract 3×A from each side of the above equation: B + 3×A – 3×A = 5×A + F – 3×A which becomes eq.4a) B = 2×A + F
Hint #7
Substitute 3×A for E, and 2×A + F for B (from eq.4a) in eq.3: A + 3×A = 2×A + F + F which becomes 4×A = 2×A + 2×F Subtract 2×A from both sides of the above equation: 4×A – 2×A = 2×A + 2×F – 2×A which becomes 2×A = 2×F Divide both sides by 2: 2×A ÷ 2 = 2×F ÷ 2 which makes A = F
Hint #8
Substitute A for F in eq.4a: B = 2×A + A which makes B = 3×A
Solution
Substitute 3×A for B and E, 5×A for C, 8×A for D, and A for F in eq.1: A + 3×A + 5×A + 8×A + 3×A + A = 21 which simplifies to 21×A = 21 Divide both sides of the above equation by 21: 21×A ÷ 21 = 21 ÷ 21 which means A = 1 making B = E = 3×A = 3 × 1 = 3 C = 5×A = 5 × 1 = 5 D = 8×A = 8 × 1 = 8 F = A = 1 and ABCDEF = 135831