Guess my number games
Guess My Number 35  Math Solutions
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A Lesson with Third, Fourth, and Fifth Graders
by Rusty Bresser and Caren Holtzman
Guess My Number invites children to consider the structure of the number system while engaging in a logic game. Students try to guess a secret number from within a given range of possibilities. Guess My Number also presents an opportunity to reinforce mathematical symbols such as the “greater than” and “less than” signs. Through this activity, students learn the usefulness of number lines as tools for solving problems. Guess My Number appears in Rusty Bresser and Caren Holtzman’s Minilessons for Math Practice, Grades 3–5 (Math Solutions Publications, 2006).
As Patty Stark’s fifth graders settled into their seats, I went to the board and drew a box with a question mark inside it.
“Good morning,” I greeted the class. “I’ve got a secret number for you to guess. Since it’s early in the morning I’m going to make it pretty easy for you. I’ll tell you the number is somewhere between one and one hundred.”
“Could it be one hundred?” Martina asked.
“Yes,” I responded. “It could be any number between one and one hundred, including one or one hundred. You can guess a number and I’ll tell you if my secret number is greater than or less than your guess.”
I drew a number line on the board to help the students keep track of their guesses.
“How about fifty?” Latoya asked.
“The secret number is less than fifty,” I told her as I wrote this information on the board. I also marked 50 on the number line with an arrow, indicating all the numbers fifty and above were too large.
Before I took any more guesses I decided to have a brief discussion about strategies for guessing. I had deliberately picked an easy number to start with so we could focus on the mechanics and thinking involved with the game.
“I’m a bit curious, Latoya,” I said. “Is there a particular reason you chose fifty?”
“Yeah,” Latoya responded. “I knew your number was between one and one hundred, so I picked fifty because it’s in the middle.”
“So how does that guess help you?” I pushed.
“Because it splits the numbers. Since you said the secret number is less than fifty, I know it’s in the bottom half of the numbers.”
I summarized Latoya’s strategy with accompanying references to the number line. “I think I get it,” I told the class. “All the possible numbers are between one and one hundred. So if you guess a number right in the middle, you can figure out which half the secret number is in and then you can just throw away the other half and not have to worry about it.”
I proceeded to take some more guesses. Within a minute the number line looked like this:
I then asked students to pair up and briefly discuss two things—what they thought they already knew about my secret number and what number they’d like to guess next. I wrote the two prompts on the board to help them stay focused. Also, I was giving them a preview of the discussion to come.
What do you already know about my secret number?
What next guess would you like to make? Why?
After a few minutes I called for the students’ attention. “So can anyone tell something you know about my secret number?” I asked.
“It’s more than forty,” Hilario offered.
“It’s between forty and fifty,” David added.
“It’s in the forties,” Brenda posited.
“So you already know a lot about the secret number,” I validated. “With all this information, what number would you like to guess next? If you raise your hand to guess a number, you also have to be willing to explain why you think that number is a helpful guess.”
I called on Destinee. “Fortyfive,” she volunteered.
“Why is fortyfive helpful?” I asked.
“Because,” Destinee explained, “we know the secret number is in the forties, and fortyfive is in the middle of the forties.”
“It’s just like Latoya’s idea,” Reynaldo chimed in. “You can get rid of half the numbers that are left. ”
“Aha,” I responded, “so you’re using your logical thinking to help eliminate a bunch of possibilities with one guess. Well, I’ll tell you that the secret number is less than fortyfive. Talk to a partner again about what you know about the number now and what guess you’d like to make next.”
I let the students talk to each other as I added the new information to the number line. Many students wanted to use the same strategy and pick the number that was halfway between 40 and 45. We took a brief detour to establish that 42.5 was the midpoint between 40 and 45, but I explained that this Guess My Number game involved whole numbers only, so they needed to choose either 42 or 43.
Christina guessed fortytwo. I told her that the secret number was greater than fortytwo and recorded this information on the board. Then Kenny guessed fortynine. Some students expressed frustration with his guess since they already knew the number was less than fortyfive. I stopped briefly to have a talk about maintaining a safe environment.
“This is a new game we’re playing today,” I told the class. “Part of learning and trying new things is making mistakes. It’s really important that everyone in the class feels safe enough to share his or her ideas and sometimes make mistakes. That’s how we learn. If you disagree with someone or you have a different idea, that’s fine. Just make sure you communicate that in a way that won’t hurt anyone’s feelings. Do you know what I’m talking about?” I asked the class as I looked at each student.
“Yes,” the students murmured.
I added Kenny’s guess to the board and moved on.
“How about fortythree?” Lisa suggested.
I wrote 43 on the board and circled it.
“Yes,” I congratulated. “My number is fortythree. I’m impressed with everyone’s thinking. It could have been any of one hundred different numbers, and it took you only eight guesses to get it. That shows you used a lot of good mathematical thinking.”
Before I left the class, I called on a pair of students to lead the class in another round of Guess My Number.
Guess My Number works equally well with fractions, decimals, or percents. Giving the students some visual tools is essential. Using a number line helps students compare numbers and order the numbers.
A 1–100 chart is another tool that works nicely for Guess My Number. Tell students that the secret number is somewhere on the 1–100 chart. Cross numbers off the chart as they are eliminated. Familiarity with a 1–100 chart gives upperelementary students a distinctive edge when it comes to mental computation and understanding our number system. When students have a visual model of the chart in their heads, they can easily jump around using tens. They also have a useful geometric model (the 10by10 square) to get a feel for how numbers are related to one another. Playing Guess My Number with a 1–100 chart gives students further exposure to the chart and pushes them to articulate some of the number relationships inherent in it.
Featured in Math Solutions Online Newsletter, Spring 2007, Issue 25
Related Publication:
Minilessons for Math Practice, Grades 35
by Jennifer M. BayWilliams and Sherri L. Martinie
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Guess the Number From Place Values Game  Math Games
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Kids often develop misconceptions about concepts in mathematics, including place value. It is important to help them get over those misconceptions. Here the rigor is beautifully balanced by asking students to work with numbers within 1,000. What better way for your second grader to learn than to have fun while learning? This game will stimulate their minds to answer a variety of questions using their understanding of place value of numbers.
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RELATED TOPICS9 useful math games for children and their parents
1. "Signals"
Age: 35 years old
Develops counting skills and the ability to concentrate.
This is a game for the little ones and it's very enjoyable because it's easy to play on the road or on the street. When you go hand in hand with a child, you simply press his palm, and he must count how many times you pressed. Then you can change.
2. "Then jump up!" nine0004
Age: 6–10 years
An especially good game when you and your child go somewhere on foot and you need to entertain them. The game is more for attention, but it also reports something about the property of numbers. The rules are simple: even  clap your hands, divided by three  jump, and if there are 7 in the number  say "Ulyul". The more signs to track, the more difficult the game.
3. "Fly"
Age: 614 years
Develops attention and spatial thinking.
To explain the game, we need a drawn 3 by 3 square. In the center of the square we need to put a dot, it will be a “trained fly”. You give the fly commands "up", "down", "right", "left", it executes them. nine0005 After you have explained the rules, you need to ask the child to imagine a field and a fly on it. Now you will order her how many cells and in which direction to move. The child must imagine these movements. If the fly went out of the field, he should clap  this means that he caught it and this is his victory. And if you didn’t slam, then the fly flew away and you won. Change roles after each game.
Changing the difficulty level. You can make the field larger or even turn it into a threedimensional one. For example, a cube 3x3x3. You can also give a series of commands without pauses. For example, down  up  right  down. nine0007
4. "Danetki" with mathematical concepts
Age: 6–12 years old
Develops categorization of concepts.
You guess any mathematical concept. For example, "circle" or "number 10". And the child asks questions to which you can answer either “yes” or “no”. There are, of course, endless possibilities in terms of difficulty.
5. "Bulls and Cows"
Age: 799 years old
Develops logical thinking
This is a number guessing game. The participants in the game think of fourdigit numbers and write them down on their sheets. All numbers included in the guessed numbers must be different from 1 to 9. The task of the players is to determine the number of the opponent by asking questions in turn. To ask a question means to name some fourdigit number, all the digits of which are different.
If the conceived and named numbers have a common figure standing in the same place, then this situation is called "Bull" (denoted by "B").
For example: the number 5739 is conceived, the opponent calls the number 1234. The number "3" in both the first and second numbers is in third place. In this case, the answer is "One bull" ("1B").
If the conceived and named numbers have the same numbers, but they are in different places, then this situation is called "Cow" (denoted by "K"). nine0005 For example: the number 5739 is conceived, the opponent calls the number 3456. The numbers "3" and "5" are the same, but are in different places. In this case, the answer is "Two
cows" ("2K").
Having heard the question, the player must carefully compare it with the planned number and correctly name the number of “bulls” and “cows”. The winner is the player who manages to guess the opponent's number in fewer questions. To avoid disputes, write down not only the opponent's answers, but also your answers to him in a separate column. nine1864 12 years
Develops logical and spatial thinking.
The playing field is a square 5 by 5 cells. Two are playing. The first puts a cross in any cell, the second puts a zero. The winner is the one who puts three in a row horizontally, vertically or diagonally. nine0005 Changing the difficulty level: there is 3D  tictactoe, and the field can also be infinitely large.
7. "Guess the number"
Age: 8–99 years old
Develops logical thinking.
I think almost everyone has played this game. You guess any number from 0 to 100 (the range can vary depending on age). The child must guess it in the least number of attempts. After each guess, you say whether your number is greater or less.
It is important that the child himself finds the most efficient way to guess  it is not very difficult (split into equal ranges, that is, 50, 25, 13, etc.). So when you switch roles, it's better to guess the number with the wrong strategy. nine0005 Difficulty change: You can agree that you have one or more opportunities to lie. The task immediately becomes much more difficult, it is worth allowing you to write down answers and reason together.
8. "Black box"
Age: 899 years old
Develops logical thinking and promotes understanding of dependencies.
You think of some function. For example, x + 6. The child tells you a number, for example 5. You substitute his number into your function: 5 + 6 = 11, and tell him the answer is 11. Then the child says another number. For example, 9, you answer him that you succeeded. In our case, it will be 9 + 6 = 15. The child must guess what you are doing with his number, that is, understand what function you have guessed. Then you can change.
Changing the difficulty level: you can think of very different functions, for example x ⋅ (x  1), and if the child is older than 67th grade, then you can build a graph using these points and at the same time see the properties of the graphs of the function.
9. Signs of divisibility
Age: 9–13 years old
It's not exactly a game, but I used to love to do it as a kid. There are interesting signs of divisibility. For example, if the difference between the sum of digits in even places and the sum of digits in odd places is divisible by 11, then the number is divisible by 11. Or, in another way, if the alternating sum of digits is divisible by 11, then the number itself is divisible by 11. Or if the sum of the digits is divisible by 9, then the whole number is divisible by 9.
I looked for numbers on the street and checked what they were divisible by. For example, I liked checking if the car number is divisible by 11. I think it is possible to offer the child to explore the world around him in this way and see if he likes it or not. Such games are easy to invent yourself. While I was writing the text, one of my colleagues came up with a similar game. nine0007
Guess 2/3 of the average, %username% / Sudo Null IT News
to 2/3 of the arithmetic mean of the numbers sent, won 5000 Danish kroner (at that time about $800).
This game is known in game theory as “guess 2/3 of the mean”. It demonstrates the difference between completely rational behavior and the real actions of the players. nine0007
Imagine that all the participants in the game act completely rationally and, no less important, know that the others also act rationally and do not collude with each other. What is the optimal number in this situation?
Obviously, it makes no sense to call numbers greater than 66.(6), because the arithmetic mean cannot be greater than 100. But, if all players reason in this way, then all the numbers will be no greater than 66.(6), which means that the arithmetic mean will not exceed this number, and therefore call more than 2/3 * 66. (6)=44.(4) again makes no sense. Repeating this argument infinitely many times, we come to the conclusion that the only correct move is the number 0. Thus, if all players reason rationally, then they must all choose the number 0.
However, the situation is different in real life. Even if the player is rational, he knows that many of his opponents are not rational, which means that he will have to take into account that their numbers will be greater than 0. We can assume that the majority will send more or less random numbers, then the average will be 50, twothirds of 50 approximately equals 33. If we go further and assume that quite a lot of people guess the number 33, then we can choose twothirds of 33, i. e. 22. Further iterations will give ~15, ~10, etc., but it seems unlikely that a sufficiently significant number of players will calculate this far. nine0007
Let's go back to the beginning of the article. What number won in Denmark? Below you see the histogram of the game, which was attended by 19196 people.
The first thing that catches your eye is the expected peaks at points 22 and 33. The winning number turned out to be a little less than 22, most likely as a result of the fact that most of the participants realized the pointlessness of choosing numbers greater than 66. (6). It is curious that there were those who sent 67 or more, including 100. I wonder if they did this without trying to win or simply did not understand the futility of such a move? It is also interesting whether those who sent 0 were guided by absolutely rational reasoning, or simply chose a round number? nine0007
Another interesting point: if in the condition of the problem the choice is limited only to integers, then there are two rational winning strategies: 0 and 1. The fact is that due to the discreteness of integers, multiplication by 2/3 cannot be repeated infinitely number of times. When we get to 1, the next iteration will give 2/3, but, rounding up to integers, we will again get 1.
I propose to play a game on Habré. Send me real numbers from 0 to 100 via habramail. an important part of the game is the players' ignorance of other people's numbers. I will announce the results when there are enough votes or a long period of time has passed. I have a theory about what the winning number will be, but I'll keep it for now :)
UPD: unregistered on Habré, send numbers to [email protected]
UPD2: participants turned out to be much more than I expected (at the moment  about 350), so the processing of the results was delayed. They will be a separate post tomorrow (already today), around the same time of day as the original post.
UPD3: the results have been processed, the winner has been determined.
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