An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Find a great variety of ways of asking questions which make 8.
Choose any three by three square of dates on a calendar page.
Circle any number on the top row, put a line through the other
numbers that are in the same row and column as your circled number.
Repeat this for a number of your choice from the second row. You
should now have just one number left on the bottom row, circle it.
Find the total for the three numbers circled. Compare this total
with the number in the centre of the square. What do you find? Can
you explain why this happens?
The closest total is 1503.
The first two correct answers to this problem
were received from Natalie and Jake, both from the West Flegg GM
Middle School. Both had arrived at the answer by trial and
Later on, work by Thomas from Wymondham High
School arrived. He had been a little more systematic in his search
and explained how he strove initially to get the hundreds column to
total 15 and the tens column to total 9. Unsuccessful at this he
moved on to consider making the units column at least 20 and the
tens column eight, which after a little searching gave him a result
that he was satisfied with.
In whatever way the digits 1 to 9 are arranged
and added together the total will always have a digital root that
e.g. 123 + 45 + 678 + 9 = 855
where 8 + 5 + 5 = 18, i.e the sum of the
and 1 + 8 = 9
123 + 456 + 789 = 1368
and the sum of the digits is 1 + 3 + 6 + 8 =
When restricted to three 3-digit numbers, they
too must have a total whose digital root is 9.
Hence, 1500 cannot be a solution because its
digital root is 6. While the integers 1494 and 1503 both have
digital roots of 9, 1503 is nearer to 1500 than 1494.
Search for the digits which would make the
units add up to 13 or 23, the tens digits add up to 9 or 8 and the
hundreds digits add up to 13 or 14 e.g. 519 + 748 +236 = 1503.
See the article Divisibility