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?
What are the missing numbers in the pyramids?
32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50
x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if
A little bit of algebra explains this 'magic'. Ask a friend to
pick 3 consecutive numbers with none greater than 60 (say for
example she picks 42, 43, 44). Ask her to tell you a multiple of 3
that is less than 100 (for example 39). Then ask her to add the
four numbers and multiply by 67, not letting you see the numbers of
course, and to tell you the last two digits of her answer. Now you
can really amaze her by giving the whole answer and the three
consecutive numbers used at the start.
This is what you do. Your friend gives you a 2 digit number and
you double it to get the remaining digits. In this case the
42+ 43 + 44 + 39 = 168, 168 x 67 = 11256
You are told the last two digits, namely 56, and you can
immediately give the whole answer 11256.
To get the three consecutive numbers you divide her multiple of
3 by 3 and add 1 (e.g. 13+1=14). Subtract this from the two digit
number she has given you and you get the first of the three
consecutive numbers in question (e.g. 56 -14 = 42).
Explain why this works.