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On the planet Vuvv there are two sorts of creatures. The Zios have $3$ legs and the Zepts have $7$ legs.
So naturally there are two forms of counting on Vuvv - Zios count in base $3$ and Zepts count in base $7$.
When observed, the creatures on this planet lie on the ground with their legs in the air, so that legs, not bodies, can be most easily counted.
One day four of these creatures, two Zios and two Zepts, sat on the summit of a hill to count the legs of the creatures they could see. One looked to the East, one to the West, one to the South and one to the North.
The creature looking to the West wrote down its number: $122$
The creature looking to the East wrote down its number: $22$
The creature looking to the South wrote down its number: $101$
The creature looking to the North wrote down its number: $41$
In which direction are the two Zios looking and in which directions are the two Zepts looking?
Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?
Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.