Fun Number Games For Children

Mathematics can be fun. The apparent paradoxes3,025, which is the original number," said Susan
of ruthless logic, the twists and tricks of simpleproudly.
arithmetic, are the delightful thorns in every"Yes, you are right," said Claire. "Let's find other
intelligent person's intellectual flesh. The truth isnumbers which are similar, and then we can tell
that we're all intellectual snobs, and games of skillthe teacher about it at the next math lesson."
and chance (where the skill is ours, the chanceSo they took pencils and paper and tried out
the other fellow's) intrigue us all, from nine tovarious numbers. Suddenly Claire exclaimed,
ninety, student to dilettante. Here is a mine of"Eureka! 9,801." Indeed, 98 + 1 ==99, and 99 x
brain-teasers and brain-trainers, to while away the99 = 9,801.
idle hour in improving your mathematical skills.A few days later at school Susan wrote down
Squares Within Squares Game: The task is tothe numbers in question on the board. "What do
write such numbers in the diagram that the sumyou think?" asked the teacher. "Are there any
of the squares of two adjacent numbers is theother numbers of this type?"
same as the sum of the squares of the two on"Please," said George, "is there a way of finding
the opposite side of the diagram. For example,such numbers without using a trial-and-error
put 16 in square A and 2 in square B. 16 2 = 256;method?" "Yes," said the teacher. "George is
2 2 = 4; 256 + 4 = 260. We said that F 2 4~ Gthinking about this just as a mathematician does
2 must be the same. Suitable numbers would bewhen he keeps trying to find a general rule to
8 and 14, because 8 2 = 64; I4 2 == 196; 64 +cover all possible solutions. Let's have a look at
196 = 260.2,025 : 20 + 25 = 45 and 45 X 45 = 2,025."
Similarly B 2 + C 2 must be equal to G 2 + H 2 ;"But our numbers are better," shouted Claire.
also, A 2 + K 2 = F 2 + E 2 ."What do you mean by better?"
What numbers must we write in the empty"Well, in our numbers all the digits are different."
squares ? Only whole numbers may be used."You are right," said the teacher. "But 2,025
Since A 2 + B 2 = F 2 + G 2 , then A 2 F 2 = Gcannot be excluded for that reason; let's see how
2 B 2 ; in other words, the difference betweenmany numbers of this type there are."
the squares of numbers on the same diagonalThey tried and tried, but apart from 3,025 (55 X
must always be the same. In our case, the55), 9,801 (99 x 99), and 2,025 (45 X 45), they
difference is i6 2 - 8 2 = I4 2 2 2 = 192could not find any others. The teacher then
Similarly, C 2 - H 2 = 192.explained that there are none.
But the difference between the squares of twoWhy? The four-figure number must be given by
numbers must be equal to the sum of thosethe square of a two-figure number; let's call this #
numbers multiplied by their difference. Using2 . Let us call the two two-digit numbers x and y.
symbols: (x y) (x -f y) = # 2 jy 2 .We are saying that the two-figure numbers are
Therefore, we can write: (C + H) (C ~ H) = 192.added, that the result is squared, and that we get
The result 192 also tells us that (C + H) and (C H)back to the original four-figure number. That is: (x
cannot both be odd numbers; otherwise, their+3 ; ) 2 = & r x + y = a an d y = a x.
product would not be even. If one (say, C + H) isAs we can see from Claire's example, we can
even, then the other must be as well, becausethink of 01 as a two-figure number, and even
the sum of the difference of the two numbersoooo is a satisfactory four-figure number.
can be even only ifboth numbers, C and H, areOn the other hand, in the original four-figure
even or if both are odd.number, x can be regarded as the number of
Expand 192, using even numbers: 2 X 96, 4 x 48,hundreds (expressing the thousands as hundreds)
6 x 32, 8 x 24, 12 X 16. Therefore: And theseand y the units (expressing the tens as units).
numbers then can be written instead of C and H.Consequently, 2 (the original number) can be
C + H = 48written as: 1oo# + y = a
C-H= 4We know that jy equals a - #; therefore,
Further: C + H = 48substituting: + a - x = a 2 = a 2 a
H = 22As we said at the beginning, x must be a whole
The Broken Board: When we insert the numbersnumber. This can only happen if a(a i) can be
into their positions, it seems as if it does notdivided by 99 without a remainder being left (99
matter which we take as I and which as H.can be expressed as 9 X n).# can be a whole
However, we must take care. If in one pair thenumber in four cases:
larger number is in the upper half of the diagram,1. a = 99 when the fraction is simplified so we get
we must be sure that the larger number is in thex = 98 and y = i, giving the four-figure number as
pair next to it in the lower half, since the sum of9,801.
the squares of two larger numbers cannot give2. a i = 99. But then a = 100, which we cannot
the same result as the sum of two smaller ones.use, as a 2 == 10,000, which is a five-figure
Continuing in this fashion, we can get the othernumber.
numbers too.3. a is divisible by 9, and (a i) by n. How do we
Susan was very interested in how numbers arefind a number like that?
related to each other. As soon as she saw aLet us write down the one and two-digit numbers
number, her imagination started working until shewhich are divisible by nine and the numbers which
found something interesting about it. "Look, Claire,"are one less than these:
she said to her friend. "Look what I have noticed.8,18,17,27,26,36,35,54,53,63,62,72,71,81,80,90, 99
Can you see that broken board?" Claire said, "Yes,and 98.
I can see it. What about it? It says 3,025."The only pair of numbers which satisfies all our
"See how two numbers were left when therequirements is 45 and 44. In this case, when we
board was broken, 30 and 25. If we add themsimplify our equation, we get: x = 20 and y = 25,
together, we get 55. And 55 X 55 (that is, 55*) isgiving 2,025 as the four-figure number.