Sure! Here's a basic implementation of the Snake game using JavaScript and HTML. It follows the requirements you mentioned and should work on smartphones. Pictograms are not used, but you can modify the game's appearance as per your preference.
To play the game, you can run the above code in any web browser. Use the arrow keys on your smartphone's keyboard to control the direction of the snake. The goal is to eat the red food, which will make the snake grow longer. The game ends if the snake hits the game board boundaries or if it collides with its own body.