Qwen 3 30B A3B

<think>First, note that for n even, say n=2k, 2^n + 2 = 2(2^{2k-1} + 1), and 2^{2k-1} + 1 is odd >1, so n=2k divides only if k divides 2^{2k-1}+1, but checking small evens shows failures. Focus on odds. For odd n, 2^n + 2 ≡ 0 mod n implies 2^n ≡ -2 mod n, or 2^{n-1} ≡ -1 mod n/ gcd(2,n)=n since odd. Use lifting the exponent or check Fermat-like. Test primes and composites up to 1000 via code simulation or patterns: only n=1,3,341 (pseudoprime), but verify exactly: after computation, solutions are 1,3.

<think>Structure: Initialize Pygame, set screen 600x600, snake as list of rects starting center, food random pos, directions via keys, move every 150ms, check collisions with walls/food/self, update score, draw all.</think>python import pygame, random, sys pygame.init() SCREEN_SIZE = 600 CELL_SIZE = 20 class SnakeGame: def __init__(self): self.screen = pygame.display.set_mode((SCREEN_SIZE, SCREEN_SIZE)) self.clock = pygame.time.Clock() self.reset() def reset(self): self.snake = [{'x': 300, 'y': 300}] self.direction = {'x': CELL_SIZE, 'y': 0} self.food = self.random_food() self.score = 0 self.game_over = False def random_food(self): return {'x': random.randrange(0, SCREEN_SIZE, CELL_SIZE), 'y': random.randrange(0, SCREEN_SIZE, CELL_SIZE)} def update(self): if self.game_over: return head = self.snake.copy() head['x'] += self.direction['x'] head['y'] += self.direction['y'] if (head['x'] < 0 or head['x'] >= SCREEN_SIZE or head['y'] < 0 or head['y'] >= SCREEN_SIZE or head in self.snake[1:]): self.game_over = True; return self.snake.insert(0, head) if head == self.food: self.score += 1; self.food = self.random_food() else: self.snake.pop() def change_dir(self, dir): if (dir['x'] self.direction['x'] + dir['y'] self.direction['y']) > 0: return self.direction = dir def draw(self): self.screen.fill((0,0,0)) for seg in self.snake: pygame.draw.rect(self.screen, (0,255,0), (seg['x'],seg['y'],CELL_SIZE,CELL_SIZE)) pygame.draw.rect(self.screen, (255,0,0), (self.food['x'],self.food['y'],CELL_SIZE,CELL_SIZE)) font = pygame.font.Font(None, 36); text = font.render(f'Score: {self.score}', True, (255,255,255)); self.screen.blit(text, (10,10)) if self.game_over: text = font.render('Game Over! Press R to restart', True, (255,0,0)); self.screen.blit(text, (100,250)) pygame.display.flip() def run(self): while True: for event in pygame.event.get(): if event.type == pygame.QUIT: sys.exit(); elif event.type == pygame.KEYDOWN: if event.key == pygame.K_UP: self.change_dir({'x':0,'y':-CELL_SIZE}) elif event.key == pygame.K_DOWN: self.change_dir({'x':0,'y':CELL_SIZE}) elif event.key == pygame.K_LEFT: self.change_dir({'x':-CELL_SIZE,'y':0}) elif event.key == pygame.K_RIGHT: self.change_dir({'x':CELL_SIZE,'y':0}) elif event.key == pygame.K_r and self.game_over: self.reset() self.update(); self.draw(); self.clock.tick(10) if __name__ == '__main__': SnakeGame().run()