Puzzle for May 13, 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.
Today, we feature another puzzle from Judah S (age 15), his 5th of this week so far! Thanks, Judah, for all of this week's puzzles!
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Hint #1
Subtract B from both sides of eq.4: A – B = B + (C × C) – B which becomes eq.4a) A – B = C × C Subtract D from both sides of eq.6: B – D = D + (C × C) – D which becomes eq.6a) B – D = C × C
Hint #2
In eq.4a, replace C × C with B – D (from eq.6a): A – B = B – D Add B and D to both sides of the above equation: A – B + B + D = B – D + B + D which becomes eq.4b) A + D = 2×B
Hint #3
Add E and B to both sides of eq.3: B – E + E + B = A – B + E + B which becomes eq.3a) 2×B = A + E In eq.3a, replace 2×B with A + D (from eq.4b): A + D = A + E Subtract A from each side of the equation above: A + D – A = A + E – A which makes D = E
Hint #4
In eq.2, substitute D for E: C = D + D which makes C = 2×D
Hint #5
Substitute 2×D for C in eq.6a: B – D = 2×D × 2×D which becomes B – D = 4×D² Add D to both sides of the above equation: B – D + D = 4×D² + D which becomes eq.6b) B = 4×D² + D
Hint #6
Substitute (4×D² + D) for B (from eq.6b) into eq.4b: A + D = 2×(4×D² + D) which becomes A + D = 8×D² + 2×D Subtract D from each side of the above equation: A + D – D = 8×D² + 2×D – D which becomes eq.4c) A = 8×D² + D
Hint #7
Substitute 8×D² + D for A (from eq.4c), and D for E in eq.5: D = 8×D² + D – (D × F) In the equation above, subtract D from both sides, and add (D × F) to both sides: D – D + (D × F) = 8×D² + D – (D × F) – D + (D × F) which simplifies to D × F = 8×D² Divide both sides by D: (D × F) ÷ D = 8×D² ÷ D which makes F = 8×D
Solution
Substitute D for E, 8×D² + D for A (from eq.4c), and 8×D for F in eq.1: D = 8×D² + D – 8×D which becomes D = 8×D² – 7×D Add 7×D to both sides of the above equation: D + 7×D = 8×D² – 7×D + 7×D which makes 8×D = 8×D² Divide both sides by 8×D: 8×D ÷ 8×D = 8×D² ÷ 8×D which makes 1 = D making A = 8×D² + D = 8×1² + 1 = 8 + 1 = 9 (from eq.4c) B = 4×D² + D = 4×1² + 1 = 4 + 1 = 5 (from eq.6b) C = 2×D = 2 × 1 = 2 E = D = 1 F = 8×D = 8 × 1 = 8 and ABCDEF = 952118