**Problem Link:** https://www.iarcs.org.in/inoi/2007/zio2007/zio2007-qpaper.pdf

**PREREQUISITES:** Basic Mathematics and good observation skills.

**The problem in short:**

We need to carefully observe the pattern and the formation of the creases namely, Valleys ( **V** ) and Hills ( **Λ** ) when we fold a sheet of paper each time in half lengthwise for a given number of times.

**Explanation:**

**----------------------------------------------Observing the formation of creases-----------------------------------------**

- When we fold and unfold the sheet of paper for the first time, a crease of single Valley(
**V**) is formed in the middle of the paper. - When we fold it twice and unfold it, we observe a pattern like
**V V Λ**. Note that the middle element**V**in this pattern, is preserved pattern from folding the paper once and two new alternative pattern**V**and**Λ**is formed. - When we fold it three times and unfold it, we observe a pattern like
**V V Λ V V Λ Λ**. Note that in this pattern the creases**V**,**V**and**Λ**in the position 2, 4 and 6 respectively, are preserved pattern from the 2nd time folding and the new creases are formed in between the 2nd time folding in an alternating pattern of**V Λ V Λ**in the position 1, 3, 5 and 7 respectively.

**Refer the image for better understanding:**

**Red arrows represent newly formed creases.****Black arrows represent middle crease formed from folding the paper for the first time.**

**(a) How many valleys are there if we fold the paper 10 times and unfold?**

**Solution:** For this question, we need to observe the number of Valleys ( **V** ) formed after each fold.

**Refer the image below for better understanding:**

**Red arrows represent newly formed creases.****Black arrows represent middle crease formed from folding the paper for the first time.**

I hope you could see the pattern already, so for 3 folds the number of Valleys (

**V**) is 2^{3-1} which is 4, so for 10 folds the number for Valleys (

**V**), would be 2^{10-1} or 2^9 or 512. In general, the number for Valleys (

**V**) would be 2^{F-1}, where F is the number of folds.

**(b) Write down the pattern of hills and valleys of the first ten creases if we fold the paper 15 times and unfold.**

**Solution:** Just observe the image below.

The pattern which appeared in the previous fold reappears in the next fold in front of the middle element, you can see the 2 fold’s pattern reappeared in front of 3 fold’s pattern and you can even see the pattern formed by 3 folds is repeated as it is in front of the 4 fold’s pattern. and so on.

So even for the 15 folds, the first 10 creases remain the same as in the first 10 creases of 4 folds which is **V V Λ V V Λ Λ V V V**.

**( c ) Write down the pattern of hills and valleys of the last ten creases if we fold the paper 18 times and unfold.**

**Solution:** For this first let us closely look at the creases formed when the paper is folded 3 times.

Pattern : **V V Λ V V Λ Λ**

Note that the 4th element is the middle element which is preserved since the first fold. Now notice the first three elements before the middle element and the last three elements after the middle element. Recognized something?. The last three elements are the reverse inverse of the first three elements. Let me tell you how.

The first three elements are **V V Λ**. Now reversing this would result in **Λ V V** and now inversing each element of this resultant would result in **V Λ Λ** which is equivalent to the last three elements. This is applicable to all creases of any number of folds.

Now if we fold the paper 18 times, the first 10 creases would be the same as the first 10 creases of 4 folds proved in (b) part. So the last ten creases of 18 folds would be the reverse inverse of the first 10 creases of the 4 folds.

First ten creases of 4 folds: **V V Λ V V Λ Λ V V V**

The reverse of it : **V V V Λ Λ V V Λ V V**

Inverse each element of the reversed pattern: **Λ Λ Λ V V Λ Λ V Λ Λ**

Required solution : **Λ Λ Λ V V Λ Λ V Λ Λ**

Remember that we’re only focusing on the first and last 10 creases of the pattern.

I hope this Helped!

Thanks!