Puzzle for June 15, 2023 ( )
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.
* "E ^ E" means "E raised to the power of E".
Scratchpad
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Hint #1
Subtract D and B from both sides of eq.1: A + D - B - D = B + F - B - D which becomes eq.1a) A - B = F - D In eq.2, subtract B from both sides, and add D to both sides: B - C - B + D = A - D - B + D which becomes -C + D = A - B which is the same as eq.2a) D - C = A - B
Hint #2
In eq.2a, replace A - B with F - D (from eq.1a): D - C = F - D Add C and D to both sides of the equation above: D - C + C + D = F - D + C + D which becomes eq.2b) 2×D = F + C
Hint #3
Add E to both sides of eq.3: C + D + E = A + B - E + E which becomes eq.3a) C + D + E = A + B
Hint #4
In eq.4, replace A + B with C + D + E (from eq.3a): E + F = C + D + E - F In the above equation, subtract D and E from both sides, and add F to both sides: E + F - D - E + F = C + D + E - F - D - E + F which simplifies to eq.4a) 2×F - D = C
Hint #5
In eq.2b, substitute 2×F - D for C (from eq.4a): 2×D = F + 2×F - D which becomes 2×D = 3×F - D Add D to both sides of the above equation: 2×D + D = 3×F - D + D which makes 3×D = 3×F Divide both sides by 3: 3×D ÷ 3 = 3×F ÷ 3 which makes D = F
Hint #6
Substitute D for F in eq.4a: 2×D - D = C which makes D = C and also makes F = D = C
Hint #7
Substitute C for D in eq.2a: C - C = A - B which becomes 0 = A - B Add B to both sides of the equation above: 0 + B = A - B + B which makes B = A
Hint #8
Substitute A for B, and C for F in eq.5: E = (A ÷ A) + (C ÷ C) which becomes E = (1) + (1) which makes E = 2
Hint #9
Substitute 2 for E in eq.6: C ÷ 2 = 2 ^ 2 which becomes C ÷ 2 = 4 Multiply both sides of the above equation by 2: 2 × (C ÷ 2) = 2 × 4 which makes C = 8 and also makes F = D = C = 8
Solution
Substitute 8 for C and D, 2 for E, and A for B in eq.3a: 8 + 8 + 2 = A + A which makes 18 = 2×A Divide both sides of the above equation by 2: 18 ÷ 2 = 2×A ÷ 2 which makes 9 = A and also makes 9 = A = B and makes ABCDEF = 998828