Derek, of St John's Primary School, sent us his working on this problem:

The lines will cross when the bottom value and the side value of the two line sare the same, and the next triangle number is bigger than the next number in the six times table. I made a table of the triangle numbers and the six times table until this happened.

Bottom value | Triangle number | Six times table |

1 | 1 | 6 |

2 | 3 | 12 |

3 | 6 | 18 |

4 | 10 | 24 |

5 | 15 | 30 |

6 | 21 | 36 |

7 | 28 | 42 |

8 | 36 | 48 |

9 | 45 | 54 |

10 | 55 | 60 |

11 | 66 | 66 |

12 | 78 | 72 |

The triangle number and the six times table number are the
same at 11. They are both 66. Then the triangle number at 12 is
bigger than the six times table number at 12, so the lines have
crossed.

I made another table to compare the square numbers with the
six times table.

Bottom value | Square number | Six times table |

1 | 1 | 6 |

2 | 4 | 12 |

3 | 9 | 18 |

4 | 16 | 24 |

5 | 25 | 30 |

6 | 36 | 36 |

7 | 49 | 42 |

The lines cross at 6. I think this is because 6 squared is 6 x
6, which is the sixth number in the six times table.

Thank you, Derek!