Puzzle for January 24, 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.
Many thanks to Judah S (age 16) for sending us this fun and interesting puzzle! Thank you, Judah!
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Hint #1
In eq.3, replace A with F - E (from eq.2): C + F = F - E + B In the above equation, subtract F from both sides, and add E to both sides: C + F - F + E = F - E + B - F + E which becomes eq.3a) C + E = B
Hint #2
In eq.1, replace B with C + E (from eq.3a): D = C + E - C which makes D = E
Hint #3
eq.5 may be written as: A + F - B = B + D + E In the above equation, substitute B - E for F - B (from eq.4), and E for D: A + B - E = B + E + E which becomes A + B - E = B + 2×E Subtract B from both sides, and add E to both sides: A + B - E - B + E = B + 2×E - B + E which makes A = 3×E
Hint #4
Substitute 3×E for A in eq.2: 3×E = F - E Add E to both sides of the above equation: 3×E + E = F - E + E which makes 4×E = F
Hint #5
Substitute 4×E for F in eq.4: B - E = 4×E - B Add E and B to both sides of the equation above: B - E + E + B = 4×E - B + E + B which makes 2×B = 5×E Divide both sides of the above equation by 2: 2×B ÷ 2 = 5×E ÷ 2 which makes B = 2½×E
Hint #6
Substitute 2½×E for B in eq.3a: C + E = 2½×E Subtract E from each side of the equation above: C + E - E = 2½×E - E which makes C = 1½×E
Hint #7
Substitute E for D, 3×E for A, and 1½×E for C in eq.6: E = 3×E ÷ 1½×E which makes E = 2
Solution
Since E = 2, then: A = 3×E = 3 × 2 = 6 B = 2½×E = 2½ × 2 = 5 C = 1½×E = 1½ × 2 = 3 D = E = 2 F = 4×E = 4 × 2 = 8 and ABCDEF = 653228