The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. Select any matchstick to create a triangle, the write out formula for pythagoreans theorem A2+B2C2, the answer will result with (C) being SQUARED. Then finally, you multiply that by 2, so 10,100 x 2 which is 20,200. So if we wanted to find out the number of toothpicks in the 100th pattern, we would do as the equation says. The list is as follow for the number of toothpicks used. Get a number of squares and make a rectangle 3 squares long by 4 squares wide. So on the 3rd pattern, you add 3+1 to get the number of horizontal rows, 4. And the amount of three toothpick squares increase by one every column. So for the first one, you have 0 "3 toothpick squares" and 1 "4 toothpick squares". The third one is made up of 3 squares: 3 toothpicks for the first and second one, and 4 for the third one. The second one is made from 2 squares: 3 toothpicks for the first square and 4 for the second. If you are trying to look for a pattern, you can see that the first column is made of 4 toothpicks. Is there a way to make a toothpick square that contains. Formula is Then you can "guess" or factor (also guessing) to get the answer. toothpicks will be in the perimeter of square 4 Write a rule that lets you predict how many. Extra Credit: Show how you can make six squares with 12 toothpicks. Show how you can make two squares with seven toothpicks (breaking or overlapping toothpicks is not allowed), three squares with 10 toothpicks, and five squares with 12 toothpicks. But when we put them back together we need to subtract 12 toothpicks, leaving us with 24. You can make one square with four toothpicks. If we separate the 9 squares in the 3rd figure, it would be 36 toothpicks. But when we put them back together, we need to remove 4, leaving the 2nd figure with only 12 toothpicks. is the number of overlapped toothpicksĪdd to get the perimeter (non-overlapping). If we separate those 4 squares, it would take 16 toothpicks to construct them. To remove overlap, note that there are perimeter toothpicks. Thus, the equation is with being the number of steps. We can see that for the case of 3 steps, there are toothpicks. Notice that the number of toothpicks can be found by adding all the horizontal and all the vertical toothpicks. The function is where is the layer and is the number of toothpicks. Using finite difference, we know that the degree must be and the leading coefficient is. We can find a function that gives us the number of toothpicks for every layer. Inspection could tell us that, so the answer is Solution 2įrom inspection, we can see that with each increase in layer the difference in toothpicks between the current layer and the previous increases by. How many steps would be in a staircase that used 180 toothpicks?Ī staircase with steps contains toothpicks. This is a 3-step staircase and uses 18 toothpicks. Step 2: Using 4 more toothpicks, poke one into the top of each gumdrop. Each gumdrop represents a vertex, which is the point where two or more sides of a shape meet. Poke the toothpicks into the gumdrops to make a square. Sara makes a staircase out of toothpicks as shown: Step 1: Start with 4 gumdrops and 4 toothpicks to make the base of your pyramid.
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