Puzzle for July 30, 2019 ( )
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.
* "A ^ B" means "A raised to the power of B".
Scratchpad
Help Area
Hint #1
In eq.4, replace E with B + C (from eq.3), and replace F with A + B (from eq.2): B + C + A + B = A + C Subtract both A and C from each side of the above equation: B + C + A + B – A – C = A + C – A – C which simplifies to 2×B = 0 which means B = 0
Hint #2
In eq.2, replace B with 0: F = A + 0 which makes F = A
Hint #3
In eq.3, replace B with 0: 0 + C = E which makes C = E
Hint #4
In eq.6, replace B with 0: A ^ 0 = C – D which means 1 = C – D (implies A ≠ 0) Add D to both sides of the above equation: 1 + D = C – D + D which means 1 + D = C and which also means eq.6a) E = C = 1 + D
Hint #5
In eq.5, substitute 1 + D for both C and E (from eq.6a): 1 + D + 1 + D = A + D which becomes 2 + 2×D = A + D Subtrtact D from both sides: 2 + 2×D – D = A + D – D which makes 2 + D = A and which also makes eq.5a) F = A = 2 + D
Solution
Substitute 2 + D for A and F, 0 for B, and 1 + D for C and E in eq.1: 2 + D + 0 + 1 + D + D + 1 + D + 2 + D = 31 which simplifies to 6 + 5×D = 31 Subtract 6 from each side of the equation above: 6 + 5×D – 6 = 31 – 6 which makes 5×D = 25 Divide both sides by 5: 5×D ÷ 5 = 25 ÷ 5 which means D = 5 making A = F = 2 + D = 2 + 5 = 7 C = E = 1 + D = 1 + 5 = 6 and ABCDEF = 706567