Applying the steps below, you will generate a number, on the condition that every digit of a number produced in each step is different from its other digits.

1. Write down a number with one, two or three digits.
2. Delete at most three consecutive digits and in the places of the deleted digits place the square of the number formed by these digits.
3. For the number you get, repeat steps two and three. If you cannot get a number that satisfies the conditions, stop.

What is the largest number that can be generated through this mechanism?

Example: 307, 3(07), 349, 3(4)9, 3169, …

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注：关于puzzleup竞赛，参见这里

We have a number. All the digits of this number are different from one another and all the digits except the first and the last have a greater value than the mean of its neighbors (ie. the digits immediately to the left and to the right).

What can this number be at maximum?

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注：关于puzzleup竞赛，参见这里

http://www.global-sci.org/mc/ 或 http://www.global-sci.org/mc/issues/

本期内容包括：

• 数学人物
• 汤涛, 姚楠: — 冯康一位杰出数学家的故事（连载二）
• 蔡天新: 冯.诺伊曼：因为他世界更加美好
• Alex Bellos: 十位伟大的数学家
• 数学趣谈
• 李尚志: 数学聊斋连载（连载二）
• 万精油: 数学与选举
• 万精油: 作家笔下的数学与数学家
• 蒋迅: 美国的数学推广月
• 数学烟云
• 罗懋康: — 後面就是秘密！ 密码漫谈 ( 续 )
• Krizek 等: 布拉格天文钟的数学原理
• 数学教育
• 张恭庆: 谈数学职业
• 数学经纬
• ukim: 聊聊数学家的故事（连载一）
• 好书推荐
• 蒋迅: 卢丁和他的《数学分析原理》
• 蔡炳坤: 我读《数学恩仇录》
• 读者来信

《数学文化》期刊由山东大学刘建亚和香港浸会大学汤涛主编；编委包括蔡天新, 张海潮, 邓明立, 顾沛，项武义, 贾朝华, 罗懋康, 张英伯, 张智民, 宗传明.
纸质版八月已经发行. 北京的九章书店可以买到; 网上订阅方式可参看
http://www.global-sci.org/mc/Subscription.html或电邮 periodical@bjzhongke.com.cn

You are going to place all numbers from 1 to 16 on a 4×4 chessboard such that all consecutive number pairs (1-2, 2-3, …, 15-16) will be on the neighboring cells (left-right-top-down).

In how many different ways can this be done?

If the question was asked for a 2×2 chessboard, the answer would be 8.

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注：关于puzzleup竞赛，参见这里

You have nine balls, three of which are slightly heavier, and three of which are slightly lighter than the normal three. All three balls in each group have the same weight. All nine balls are identical otherwise. Your mission is to determine the group of every ball using a pair of scales.

How many weighings are needed in order to ensure the accomplishment of your mission?

Notes:
1. The difference of weight between the heavy and normal balls is not necessarily equal to the difference of weight between the light and normal balls.
2. The scales show only which side is heavier; it does not show the difference of weight.

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注：关于puzzleup竞赛，参见这里

What is the largest number that cannot be expressed as a sum of different square numbers?

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Which is the smallest number that can be expressed as the sum of six different square numbers, but cannot be expressed as the sum of five different square numbers?

(Square numbers: 0, 1, 4, 9, 16, 25, …)

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A group of people is sitting around a round table. They give a coffee break. When they return to the table after the break, they sit down randomly. Interestingly, they notice that the six closest persons sitting next to each one of them (three to the left, and three to the right) are completely different from the six closest persons in the previous setting.

At least how many persons should be there?

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注：关于puzzleup竞赛，参见这里

You are using a drawing program on a computer. You place several equilateral triangles of the same size on the screen. You observe that you can cover any of these triangles by moving the other triangles without rotating.

What is the minimum number of triangles you have to place on the screen to ensure that this observation holds true for every case?

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注：关于puzzleup竞赛，参见这里