Puzzle for November 27, 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.
* DE and BC are 2-digit numbers (not D×E or B×C).
** "D ^ C" means "D raised to the power of C".
Scratchpad
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Hint #1
eq.4 may be written as: E = (A + B + D) ÷ 3 Multiply both sides of the equation above by 3: 3 × E = 3 × (A + B + D) ÷ 3 which becomes eq.4a) 3×E = A + B + D
Hint #2
Add A to both sides of eq.2: A + E + F = A + B + D In the above equation, replace A + B + D with 3×E (from eq.4a): A + E + F = 3×E Subtract E from each side: A + E + F – E = 3×E – E which becomes eq.2a) A + F = 2×E
Hint #3
Add F to both sides of eq.3: B – F + F = A – E + F which becomes eq.3a) B = A + F – E In eq.3a, replace A + F with 2×E (from eq.2a): B = 2×E – E which makes B = E
Hint #4
In eq.2, substitute B for E: B + F = B + D Subtract B from both sides of the above equation: B + F – B = B + D – B which makes F = D
Hint #5
Substitute F for D, and A + F – E for B (from eq.3a) in eq.1: A + C + F = A + F – E + E which becomes A + C + F = A + F Subtract A and F from both sides of the equation above: A + C + F – A – F = A + F – A – F which makes C = 0
Hint #6
eq.5 may be written as: A + 10×D + E = 10×B + C + F Substitute F for D, E for B, and 0 for C in the equation above: A + 10×F + E = 10×E + 0 + F Subtract E and F from each side: A + 10×F + E – E – F = 10×E + 0 + F – E – F which becomes eq.5a) A + 9×F = 9×E
Hint #7
Subtract F from each side of eq.2a: A + F – F = 2×E – F which becomes A = 2×E – F Substitute 2×E – F for A in eq.5a: 2×E – F + 9×F = 9×E which becomes 2×E + 8×F = 9×E Subtract 2×E from both sides of the above equation: 2×E + 8×F – 2×E = 9×E – 2×E which makes 8×F = 7×E Divide both sides by 8: 8×F ÷ 8 = 7×E ÷ 8 which makes F = ⅞×E and also makes D = F = ⅞×E
Hint #8
Substitute ⅞×E for F in eq.2a: A + ⅞×E = 2×E Subtract ⅞×E from each side of the equation above: A + ⅞×E – ⅞×E = 2×E – ⅞×E which makes A = 1⅛×E
Solution
Substitute 0 for C, 1⅛×E for A, and E for B in eq.6: D ^ 0 = 1⅛×E – E which means 1 = ⅛×E (assumes D ≠ 0) Multiply both sides of the equation above by 8: 8 × 1 = 8 × ⅛×E which makes 8 = E making A = 1⅛×E = 1⅛ × 8 = 9 B = E = 8 D = F = ⅞×E = ⅞ × 8 = 7 and ABCDEF = 980787