Puzzle for February 17, 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
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Hint #1
Add D to both sides of eq.4: D – E + F + D = A – D + E + D which becomes eq.4a) 2×D – E + F = A + E In eq.2, replace A + E with 2×D – E + F (from eq.4a): B + D – F = 2×D – E + F In the equation above, subtract D from both sides, and add F to both sides: B + D – F – D + F = 2×D – E + F – D + F which simplifies to eq.2a) B = D – E + 2×F
Hint #2
Add F and E to both sides of eq.6: F – E + E + F = C – F + E + F which becomes eq.6a) 2×F = C + E In eq.2a, replace 2×F with C + E (from eq.6a): B = D – E + C + E which becomes eq.2b) B = D + C
Hint #3
In eq.3, substitute D + C for B (from eq.2b): C + D = D + C – D which becomes C + D = C Subtract C from both sides of the above equation: C + D – C = C – C which makes D = 0
Hint #4
Substitute 0 for D in eq.2b: B = 0 + C which makes B = C
Hint #5
Substitute 0 for D in eq.5: E – A = A – 0 which becomes E – A = A Add A to both sides of the above equation: E – A + A = A + A which makes E = 2×A
Hint #6
Substitute 0 for D, and 2×A for E in eq.4: 0 – 2×A + F = A – 0 + 2×A which becomes –2×A + F = 3×A Add 2×A to both sides of the above equation: –2×A + F + 2×A = 3×A + 2×A which makes F = 5×A
Hint #7
Substitute 0 for D, 5×A for F, and 2×A for E in eq.2: B + 0 – 5×A = A + 2×A which becomes B – 5×A = 3×A Add 5×A to both sides of the above equation: B – 5×A + 5×A = 3×A + 5×A which makes B = 8×A and also makes C = B = 8×A
Solution
Substitute 8×A for B and C, 0 for D, 2×A for E, and 5×A for F in eq.1: A + 8×A + 8×A + 0 + 2×A + 5×A = 24 which simplifies to 24×A = 24 Divide both sides of the equation above by 24: 24×A ÷ 24 = 24 ÷ 24 which means A = 1 making B = C = 8×A = 8 × 1 = 8 E = 2×A = 2 × 1 = 2 F = 5×A = 5 × 1 = 5 and ABCDEF = 188025