Find the pattern for these equations, and then make up one more of your own, using the same pattern.

12 � 42 = 21 � 24

13 � 62 = 31 � 26

13 � 62 = 31 � 26

We observe: | 12 x 42 = | 21 | x | 24 |

is 12 reversed |
is 42 reversed |

We try, say, 12 x 52 ≟ 21 x 25 → 624 ≠ 525. No good, we need to know more.

We observe: 12 x 42 = 21 x 24 and 13 x 62 = 31 x 26

We try 14 x 82 ≟ 41 x 28 → 1148 = 1148 Ah!

Algebraically:

1 | 2 | x | 4 | 2 | = | 2 | 1 | x | 2 | 4 |

↑ | ↑ | ↑ | ↑ | ↑ | ↑ | ↑ | ↑ | |||

t |
u |
w |
u |
u |
t |
u |
w |

→ (10*t* + *u*)(10*w* + *u*) = (10*u* + *t*)(10*u* + *w*)

→ 100*tw* + 10*tu* + 10*uw* + *u*^{2} = 100*u*^{2} + 10*uw* + 10*tu* + *tw*

→ 100*tw* + *u*^{2} = 100*u*^{2} + *tw*

→ 99*tw* = 99*u*^{2}

→ *tw* = *u*^{2}.

Try *t* = 8, *w* = 2, *u* = 4

84 x 24 ≟ 48 x 42 → 2016 = 2016.

Notice that 13 x 62 = 31 x 26 doesn’t quite follow this rule. What is the most general form of the rule?