Puzzle for May 12, 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 positive integer.
* "E ^ C" means "E raised to the power of C".
Once again, we thank Judah S (age 15) for sending us another interesting puzzle. Thank you, Judah, for all of this week's puzzles!
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Hint #1
Subtract A from each side of eq.3: B – A = A + C + D – A which becomes B – A = C + D In the above equation, replace B – A with F (from eq.2): eq.3a) F = C + D
Hint #2
In eq.1, replace F with C + D (from eq.3a): E = C + D – D which makes E = C
Hint #3
In eq.4, substitute D – (C × C) for A (from eq.5), and C for E: D = D – (C × C) + C + C which becomes D = D – (C × C) + 2×C In the above equation, subtract D from both sides, and add (C × C) to both sides: D – D + (C × C) = D – (C × C) + 2×C – D + (C × C) which simplifies to (C × C) = 2×C Divide both sides by C: (C × C) ÷ C = 2×C ÷ C which makes C = 2 and also makes E = C = 2
Hint #4
In eq.6, substitute 2 for both C and E: B = (2 × 2) + (2 ^ 2) which becomes B = 4 + 4 which makes B = 8
Hint #5
Substitute 2 for C in eq.3a: eq.3b) F = 2 + D Substitute 8 for B in eq.2: eq.2a) F = 8 – A
Hint #6
In eq.3b, substitute 8 – A for F (from eq.2a): 8 – A = 2 + D Subtract 2 from each side of the equation above: 8 – A – 2 = 2 + D – 2 which becomes eq.3c) 6 – A = D
Hint #7
Substitute 6 – A for D (from eq.3c), and 2 for C and E in eq.4: 6 – A = A + 2 + 2 which becomes 6 – A = A + 4 In the above equation, add A to both sides, and subtract 4 from both sides: 6 – A + A – 4 = A + 4 + A – 4 which makes 2 = 2×A Divide both sides by 2: 2 ÷ 2 = 2×A ÷ 2 which makes 1 = A
Hint #8
Substitute 1 for A in eq.3c: 6 – 1 = D which makes 5 = D
Solution
Substitute 5 for D in eq.3b: F = 2 + 5 which makes F = 7 and ABCDEF = 182527