536 Puzzles and Curious Problems

Miscellaneous Puzzles 73

237. SHORT CUTS

We have from time to time given various short cuts in mental arithmetic.
Here is an example that will interest those who are unfamiliar with the process.
Can you multiply 993 by 879 mentally?
It is remarkable that any two numbers of two figures each, where the tens
are the same, and the sum of the units digits make 10, can always be multi-
plied mentally thus: 97 X 93 = 9,021. Multiply the 7 by 3 and set it down,
then add 1 to the 9 and multiply by the other 9, 9 X 10 = 90.
This is very useful for squaring any number ending in 5, as 852 = 7,225.
With two fractions, when we have the whole numbers the same and the sum
of the fractions equal unity, we get an easy rule for multiplying them. Take
7¥.! X 73 /4 = 5M16. Multiply the fractions = 1'16, add 1 to one of the 7's, and
multiply by the other, 7 X 8 = 56.

238. MORE CURIOUS MULTIPLICATION

Here is Professor Rackbrane's latest:

What number is it that, when multiplied by 18,27,36,45,54,63,72,81, or

99, gives a product in which the first and last figures are the same as those in

the multiplier, but which when multiplied by 90 gives a product in which the

last two figures are the same as those in the multiplier?

239. CROSS-NUMBER PUZZLE

On the following page is a cross-number puzzle on lines similar to those of

the familiar cross-word puzzle. The puzzle is to place numbers in the spaces

across and down, so as to satisfy the following conditions:

Across-I. a square number; 4. a square number; 5. a square number;

8. the digits sum to 35; 11. square root of 39 across; 13. a square number; 14.
   a square number; 15. square of 36 across; 17. square of half 11 across;


9. three similar figures; 19. product of 4 across and 33 across; 21. a square
   number; 22. five times 5 across; 23. all digits alike, except the central one;


10. square of 2 down; 27. see 20 down; 28. a fourth power; 29. sum of 18
    across and 31 across; 31. a triangular number; 33. one more than 4 times 36
    across; 34. digits sum to 18, and the three middle numbers are 3; 36. an odd
    number; 37. all digits even, except one, and their sum is 29; 39. a fourth power;


11. a cube number; 41. twice a square.