Matchsticks

5、There are ten matchsticks lined up in 4 rows to form a shape as below. Only 3 matchsticks can be moved so that the shape becomes upside down.


Which 3 matchsticks should be moved?


Write down the number of the matchsticks in the squares of the new shape

Share oranges

3、serveral children are sharing some oranges, If each child takes 1 orange, there will be 1 orange remaining. However, if each child takes 2 oranges, then the last child will get nothin. How many children and oranges are there?
1) with equation:
Let n be the number of children:
n+1 = 2 (n -1)
n = 3

2)
First, give each child 1 orange. there 1 remaining, then give it to the first child.
Take the orange of the last child and give to the second child
Take the orange of the second last child and give to the third child....
so counting forwar, one more than half of children have 2 oranges, and the others have none. but there only one child got none. So the number of children is 2×1+1 = 3.

Colored 3×3 cube

3、A large can of red paint fell on this 3×3 cube, and all its surfaces, including its bottom, got covered with red paint.

1) The number of unit cubes have only one face painted red is 6 which equals to the faces of a cube.
2) The number of unit cubes have only two faces painted red is 12 which equals to the edges of a cube.
3）The number of unit cubes have only three faces painted red is 8 which equals to the vertexes of a cube.
4) The number of unit cubes have no face painted red is 1 which is the center of the larger cube.

數三角

8、How many small triangles are there?

1、 The number of small triangles: 12

2、 The number of medium triangles: 6

3、 The number of large triangles: 2

4、 The number of triangles: 20

Find the numbers

7、A Teacher thinks of two consecutive numbers in the range 1 to 10, and tells Alex one of the numbers and Sam the other.
Sam and Alex have the following conversations:
Alex: I don't know your number.
Sam: I don't know your number, either.
Alex: Now I know!

Can you find all 4 solutions?

因為兩人數是相鄰的，所以要知道另外一個人的號碼，自己必須持有1或者10。
因此Alex持有的可能是9，因為當他持有9的時候，Sam只可能是8或者10，而Sam又不能確定Alex的號碼，所以他不可能是10或者1，所以Alex可以知道Sam持有的是8。
同樣的道理：Alex持有的可能是2，Sam是3。

直式重現

2、Find the values (0~9) of the lettres. (use each number once only).

   B F 4
+  2 8 D
 A C G E

1、因為A必定是後面的進位，而兩個數相加，進位必定為1。所以 A = 1。

2、因為B+2要產生進位，所以B可能為7（F+8必須有進位）、8、9中的一個，而F不可能是1或 者2，所以F+8有進位。如果B是8，則C會等于1，和A重復，所以B不是8；如果B是9，則C會等于2，而2已經使用。所以B = 7

3、F+8有進位，所以C = 0

4、D不能等于0，1，2，3，4，6，7，8。只能是5或者9。如果D等于5，那么F和G，只能在6和9中選擇，顯然都不可能。所以D = 9

5、F和G在5，6中選擇，容易知道F = 6 ； G = 5

所以：

A = 1

B = 7

C = 0

D = 9

E = 3

F = 6

G = 5

Magic Square

1、Write down a number in each of the following square using numbers 1 to 16 (use each number ince only) so that the sum of the 4 numbers in each row, column and diagonal is eauql to 34

行程問題001

兄妹兩人同時離家去上學。哥哥每分鐘走90米，妹妹每分鐘走60米，哥哥到校門時發現忘記帶作業本，立即沿原路回家去取，行至離學校180米處和妹妹相遇。他們家和學校的距離是多少米？

方法一：

方法二：

花圃的周界

下面是一個曲線型花圃設計時使用的圖，其中黑點表示圓心，而每個圓的半徑都是相同的，半徑長3M。求這個花圃的周界（圖中的直線是設計時的輔助線，實際建筑中并不存在）。

巧算001

計算下面的式子：

1993+1992-1991-1990+1989+1988-1987-1986+...+9+8-7-6+5+4-3-2+1=()

提示：找出數列的規律，利用加法的結合律來計算。