Puzzle for January 7, 2020 ( )
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.5, substitute (A + F) for D (from eq.4): C – (A + F) = A – F which is equivalent to C – A – F = A – F Add A and F to both sides of the above equation: C – A – F + A + F = A – F + A + F which simplifies to C = 2×A
Hint #2
Add D and F to both sides of eq.5: C – D + D + F = A – F + D + F which becomes eq.5a) C + F = A + D eq.2 may be written as: A + B + D = C + F + E In the equation above, replace C + F with A + D (from eq.5a): A + B + D = A + D + E Subtract both A and D from each side of the equation above: A + B + D – A – D = A + D + E – A – D which simplifies to B = E
Hint #3
Add E to both sides of eq.6: B + E + E = D – E + E which becomes B + 2×E = D Replace E with B: B + 2×B = D which makes eq.6a) 3×B = D
Hint #4
Substitute C for E + F (from eq.3) in eq.2: eq.2a) A + B + D = C + C
Hint #5
Substitute 3×B for D, and 2×A for C in eq.2a: A + B + 3×B = 2×A + 2×A which becomes A + 4×B = 4×A Subtract A from each side: A + 4×B – A = 4×A – A which makes 4×B = 3×A Divide both sides by 4: 4×B ÷ 4 = 3×A ÷ 4 which means B = ¾×A and also means E = B = ¾×A
Hint #6
Substitute (¾×A) for B in eq.6a: 3×(¾×A) = D which makes 2¼×A = D
Hint #7
Substitute 2¼×A for D in eq.4: 2¼×A = A + F Subtract A from both sides of the above equation: 2¼×A – A = A + F – A which makes 1¼×A = F
Solution
Substitute ¾×A for B and E, 2×A for C, 2¼×A for D, and 1¼×A for F in eq.1: A + ¾×A + 2×A + 2¼×A + ¾×A + 1¼×A = 32 which simplifies to 8×A = 32 Divide both sides of the above equation by 8: 8×A ÷ 8 = 32 ÷ 8 which means A = 4 making B = E = ¾×A = ¾ × 4 = 3 C = 2×A = 2 × 4 = 8 D = 2¼×A = 2¼ × 4 = 9 F = 1¼×A = 1¼ × 4 = 5 and ABCDEF = 438935