To demonstrate a bit of geometry and trigonometry to my grandson, I took a rectangular piece of paper whose shorter sides were 24 cm in length. With one straight fold I brought one corner of the rectangle to the midpoint of the opposite longer side. Then I cut the paper along the fold, creating a triangle and another piece. I then demonstrated to my grandson that this other piece had double the area of the triangle.

How long was the cut?

]]>At tennis a set is won by the first player to win 6 games, except that if it goes to 5 games all it is won either 7 games to 5 or 7 games to 6. (As far as this puzzle is concerned this applies even to the final set).

The match that we are considering went to 5 sets and no two sets contained the same number of games. At the end of each set the total number of games played up to that point was always a prime number. From this information the score in one or more of the five sets can be deduced with certainty.

Which set or sets had a score that can be deduced with certainty, and what was the score in the set or each of the sets concerned?

]]>The quiet town of Spirechester is divided into nine square parishes as shown on the map. Each parish church has its spire precisely at the centre of the parish, and these are marked on the map.

The churches are named after St Agnes, St Brigid, St Cecilia, St Donwen, St Etheldreda, St Felicity, St Genevieve, St Helen and St Isabel. Each spire is topped by a weather vane which has, instead of a cock, the initial letter of its saint’s name.

Recently I walked in the meadows which surround the town and took a number of photos from different positions. Fortunately, no spire was ever hidden by another spire and the wind was such that the weather vane letters were clearly visible. However, I was not sufficiently distant from the town to capture all nine spires and, in fact, each photo contains just five spires. The orders of the spires of the photos, reading from left to right, were GEIAC, EACHD, EACDH, AECHD, IFCGB.

Starting with A (for St Agnes) list the eight churches in clockwise order around the outside of the square.

]]>George has 27 small blocks which have been identified with 27 different prime numbers — each block has its number on each face. He has assembled the blocks into a 3 x 3 x 3 cube,. On each of the three visible faces, the nine numbers total 320 — but this is not true of the three hidden faces.

George remembers that when he bought the blocks they were assembled into a similar 3×3×3 cube, but on that occasion they showed the same total on each of the six faces, this being the smallest possible total if each block has a different prime number.

What was the total on each face when George bought the blocks?

]]>Thirteen toy tiles comprised a square, rectangles, rhombuses (diamonds on a playing card are rhombuses) and kites (as shown in the diagram). All of each different type were identical. A rhombus’s longer diagonal was a whole number of inches (equalling any diagonal of any other type). Its shorter diagonal was half this. Also, one side of a rectangle was slightly over one inch.

A pattern I made, using every tile laid flat, had all the symmetries of a square. After laying the first tile, each subsequent tile touched at least one other previously placed tile. Ultimately, any contact points were only where a vertex of a tile touched a vertex of just one other tile; only rhombuses touched every other tile type.

What, in inches, was a square’s diagonal?

]]>**__T H R E E**

**+ T H O U S
+ A N D T H
===========
T E A S E R
===========**

In this addition digits have been consistently replaced by letters, different letters representing different digits. But instead of an addition in base 10 in which the letters represent the digits 0 to 9 this is an addition in base 11, using X for the extra digit, in which the letters represent the digits 1 to X, with 0 unused.

Please submit the number (still in base 11) represented by **TEASER**.

“It takes all sorts to make a hang-gliders’ convention go with a whirl,” confided Icarus Thorns to the Rev E. B. Inept as they dallied over their tea on a mountain top.

“You can say that again,” answered his companion, dipping into the bag of proffered sweets and adding absent-mindedly another two lumps to his already sweetened tea. “You can say that again,” echoed the mountains faintly.

“This puzzle, unlike our jump, needs little preamble. In the (correct) equation:

**ABC – D – E – F – G – H – I – J = 100**

each of the letters stands for a different digit. All you have to do is deduce the value of:

**(D × E × F × G × H × I × J) ÷ ABC**

So saying, the intrepid Inept threw himself down off the outcrop, his wings glinting in the sun.

Can you deduce the answer before he reaches the bottom?

]]>For the purposes of this Enigma I shall define a “palindromic angle” as one whose n umber of degrees is a whole number less than 180 and is such that the number remains the same if its digits (or digit) are written in reverse order.

I have drawn an irregular hexagon and a line across it in a similar fashion to the one shown here, however the given picture is not to scale. In my figure, the eight marked angles are different palindromic angles, and in fact more than two of them are acute.

What (in increasing order) are the six angles of my hexagon?

]]>Take a large sheet of lined paper. On line 1 write any number between 0 and 100 (but not necessarily a whole number). In turn, write a number on each of the lines, 2, 3, 4, .. 100, according to the following rule:

Suppose you have just written a number X on a line. If X is less than 50 then write 40 plus half of X on the next line, otherwise write 93 minus half of X.

(A) Can I say now, before you write your number on the 1st line, what the nearest whole number to the number you write on the 100th line will be? If yes, then what is that nearest whole number?

Take a second sheet of lined paper and repeat the above, except that the rule if X is less than 50 is changed; now write 20 plus half of X on the next line.

(B) My question now is as in question A.

Now take a third sheet of lined paper and repeat the procedure, except that the rule becomes the following. If X is less than 50 then write 40 plus half of X, otherwise, write 15 plus half of X.

(C) Once again, my question is as in question A.

]]>Robbie leans two very thin playing cards together, then another two together, placing an identical card across the top forming a platform, and proceeding sideways and upwards to build a roughly triangular tower.

For the bottom layers, he uses a whole number of 53-card packs of large cards (integer length above 70mm), the number of packs equalling the number of bottom layers. He then uses small cards (75% size) to complete the tower, which is 1428mm high. The distance between the bases of two leaning cards is always 0.56 of the length of each card.

Robbie would like to extend the tower sideways and upwards to the next possible integer height, still using large cards only for the bottom layers.

How many extra cards would be needed in total?

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