Crack the code

Sanders publishing is not the largest publisher in the Netherlands of puzzles, but it is a publisher with an eye for innovation. And risk. Consider the following puzzles:

1) Crack the code 1
F + A = 6
B + C = 9
C + D = 10
D + E = 11
A + B = 10
E + F + D = 12

Each of the letters A-F stands for one of the numbers 0-6. Several letters may have the same value.

Of course for mathematicians, this puzzle is a set of 6 equations with 6 unknowns. Algebra has the reputation to be very unpopular, so it surprises me that the publisher has already published 5 issues of this magazine.

The limited range of the numbers allows for fewer equations, and here are two examples.

2) Crack the code 2
C x F = 0
E + F = 11
A x B = 12
D x E = 5
B + F = 10

3) Crack the code 3
A + D = 11
E + B = 2
B x C = 6
F * D = 20
E + C = 3

You can find the solutions at 235, 245 and 86.

This is the last post of 2012. 2012 enabled me to publish over 35 posts in this blog, with about 50 puzzles. It looks like, health and wealth permitting, I will be able to continue a weekly frequency in 2013. I look forward to your visits in 2013. 🙂

Coffee with milk, please

Tanya Khovanova publishes an irregular but excellent blog about math problems. Of Russian descent, she often uses Russian sources, which are otherwise not very accessible in the Western world. The next problem comes from her blog, and has the Moscow 2011 mathematics olympiad as origin:

1) Coffee with milk, please^***
1) Coffee and milk^**
In a certain family everyone likes their coffee with milk. At breakfast everyone had a full cup of coffee. Given that Alex consumed a quarter of all consumed milk and one sixth of all coffee, how many people are there in the family?

The above problem would go into the class of problems for which you have n equations and n+1 unknowns. Here’s a classic in this category:

2) A farmer went to the market^*
A farmer buys 100 animals for 100 dollars but lost his receipt. Cows are $10 each, pigs are $3 each and chicks are $.50 each. How many of each did he buy?
This puzzle is a ‘classic’, but I don’t know its source. If you do, I’d welcome this information!

When you solved both, you will notice that the solving methods of the two puzzles are totally different.

You can find hints at 126, 116 respectively.

Triangle sums

Each cell in this triangle is the sum of the two cells below it. Can you complete them?
1) triangle 1^*

				75
			43		.
		28		.		.
	18		.		.		.
.		.		.		.		8

I first encountered this type of puzzle in the Dutch translation of “One minute puzzles”, published by Arcturus Publishing Limited, London. This book published the numbers in circles, and all puzzles had the difficulty level of the one above, where there is always at least one cell which can be calculated with a simple addition or subtraction.
I replaced the circles with the pyramid pictured above, and in the following puzzles you will find an extra difficulty level introduced.

2) triangle 2^*

				.
			.		.
		18		.		15
	.		.		.		6
7		.		.		.		2

3) triangle 3^*

				115
			.		.
		19		.		36
	.		.		.		.
7		.		.		.		8

4) triangle 4^*

				80
			.		.
		18		.		24
	.		.		.		.
.		1		.		4		.

This type of puzzles exercises the parts of your brain which performs the arithmetic. If I interpret this article correctly, that is the horizontal segment of the bilateral intraparietal sulcus (HIPS), together with the precentral sulcus and inferior frontal gyrus.

You can check your solutions at solution 221, solution 231, solution 241, and solution 212.