Puzzle for August 3, 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
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Hint #1
eq.5 may be written as: C – F = A + F + D In the equation above, replace A + F with C – E (from eq.3): C – F = C – E + D Subtract C from both sides, and add F and E to both sides: C – F – C + F + E = C – E + D – C + F + E which simplifies to eq.5a) E = D + F
Hint #2
In eq.2, replace E with D + F (from eq.5a): D + F + F = A + D which becomes D + 2×F = A + D Subtract D from each side of the equation above: D + 2×F – D = A + D – D which makes 2×F = A
Hint #3
Subtract C from each side of eq.6: B – F – C = C + D + E – C which becomes B – C – F = D + E In the above equation, substitute A + E for B – C (from eq.4): A + E – F = D + E Subtract E from each side of the equation above: A + E – F – E = D + E – E which becomes eq.6a) A – F = D
Hint #4
Substitute 2×F for A in eq.6a: 2×F – F = D which makes F = D
Hint #5
Substitute F for D in eq.5a: E = F + F which makes E = 2×F
Hint #6
Substitute 2×F for E and A in eq.3: C – 2×F = 2×F + F which becomes C – 2×F = 3×F Add 2×F to both sides of the equation above: C – 2×F + 2×F = 3×F + 2×F which makes C = 5×F
Hint #7
Substitute 5×F for C, and 2×F for A and E in eq.4: B – 5×F = 2×F + 2×F which becomes B – 5×F = 4×F Add 5×F to both sides of the above equation: B – 5×F + 5×F = 4×F + 5×F which makes B = 9×F
Solution
Substitute 2×F for A and E, 9×F for B, 5×F for C, and F for D in eq.1: 2×F + 9×F + 5×F + F + 2×F + F = 20 which simplifies to 20×F = 20 Divide both sides of the above equation by 20: 20×F ÷ 20 = 20 ÷ 20 which means F = 1 making A = E = 2×F = 2×1 = 2 B = 9×F = 9×1 = 9 C = 5×F = 5×1 = 5 D = F = 1 and ABCDEF = 295121