快猫短视频

#214 Seven up!

Set by Mary Ellis

My burglar alarm won鈥檛 stop beeping. I need to enter the four-digit code, but I just can鈥檛 remember what it is.

I do remember that I chose a number that is divisible by 7. I also know that I chose a number in the thousands. I recall that the numbers formed by the first three digits and the last three are divisible by 7 as well. I also deliberately chose four different digits. Finally, I know that if I add all the digits, their sum isn鈥檛 divisible by 7, but if I add the digits of that sum, the result is divisible by 7.

Can you help me work out what my code is to stop the incessant beeping?

Solution next week

#213 Cross sward

Solution

The red dashed line halves the lawn, leaving the vegetable plot intact; the black dashed line halves the lawn and the plot.

Any straight line going through the centre of a rectangle will divide it in half, whatever the line鈥檚 angle, so the lawn can be halved by splitting it into two rectangles, then drawing a line joining their centres. If the partition passes through the centre of the garden and the centre of the plot, the plot and the whole garden will be split exactly in half. If this is the case, it follows that the lawn must also be split into two equal areas.