Puzzle for July 29, 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
Add E to both sides of eq.2: B + E + E = A – E + E which becomes eq.2a) B + 2×E = A In eq.5, replace A with B + 2×E (from eq.2a): B – F = B + 2×E – B – E which becomes B – F = E Add F to both sides of the above equation: B – F + F = E + F which becomes eq.5a) B = E + F
Hint #2
In eq.4, replace B with E + F (from eq.5a): C + F = A + E + F – D In the equation above, subtract F from both sides, and add D to both sides: C + F – F + D = A + E + F – D – F + D which becomes eq.4a) C + D = A + E
Hint #3
In eq.4a, substitute D + F for A + E (from eq.3): C + D = D + F Subtract D from both sides of the above equation: C + D – D = D + F – D which makes C = F
Hint #4
eq.6 may be written as: C = (A + D + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × C = 3 × (A + D + E) ÷ 3 which becomes 3×C = A + D + E which is the same as eq.6a) 3×C = A + E + D
Hint #5
Substitute C + D for A + E (from eq.4a) in eq.6a: 3×C = C + D + D which becomes 3×C = C + 2×D Subtract C from each side of the above equation: 3×C – C = C + 2×D – C which makes 2×C = 2×D Divide both sides by 2: 2×C ÷ 2 = 2×D ÷ 2 which makes C = D and also makes F = C = D
Hint #6
Substitute F for C and D, and B + 2×E for A (from eq.2a) in eq.6a: 3×F = B + 2×E + E + F which becomes 3×F = B + 3×E + F Subtract F from each side of the equation above: 3×F – F = B + 3×E + F – F which becomes eq.6b) 2×F = B + 3×E
Hint #7
Substitute E + F for B (from eq.5a) into eq.6b: 2×F = E + F + 3×E which becomes 2×F = F + 4×E Subtract F from each side of the equation above: 2×F – F = F + 4×E – F which makes F = 4×E and also makes C = D = F = 4×E
Hint #8
Substitute 4×E for F in eq.5a: B = E + 4×E which makes B = 5×E
Hint #9
Substitute 5×E for B in eq.2a: 5×E + 2×E = A which makes 7×E = A
Solution
Substitute 7×E for A, 5×E for B, and 4×E for C and D and F in eq.1: 7×E + 5×E + 4×E + 4×E + E + 4×E = 25 which simplifies to 25×E = 25 Divide both sides of the above equation by 25: 25×E ÷ 25 = 25 ÷ 25 which means E = 1 making A = 7×E = 7 × 1 = 7 B = 5×E = 5 × 1 = 5 C = D = F = 4×E = 4 × 1 = 4 and ABCDEF = 754414